I know that gentiles must follow the 7 Noahide laws. But are there any other moral principles branching out from these that are highly recommended for everyone to follow?
In totality, when completely expanded out from the seven laws: How many rules are acceptable for gentiles to follow, on the basis of morality and ethics? And what are these rules?
Please provide a list, if possible.
Several words in your question need unpacking: seven, must follow, moral principles, branch out and recommended. I will try to do a bit for each one to help answer the question, but each one could really be a whole book.
Even the number seven (and what that "seven" includes) is subject to a dispute in the Talmud (Sanhedrin 56). Thus one of the earliest talmudic commentators, R. Nissim b. Yaakov of Kairouan (or "Rav Nissim Gaon", 11th century), writes in the introduction to his Sefer ha-Mafteach le-Man'ulei ha-Talmud:
מצינו שאדם הראשון הטען מהם קצת מצוות כמו שאמרו רבותינו ז"ל שבת מצוות נצטוו בני נח, דינים וברכת השם וע"ז וגילוי עריות ושפיכות דמים וגזל ואבר מהחי, ואמרינן, מנהני מילי, אמר ר' יוחנן דאמר קרא "ויצו ה' אלהים על האדם"... וכבר הוסיף עליהן ר' חנניה בן גמליאל אף הדם מן החי… והיו מיתוספים והולכים עת אחר עת עד שעלה חשבון המצות של שמועה כ"ח ויש שאמרו כי ל' מצות הן שנצטוו בהן קודם מתן תורה
We find that God "loaded" Adam with some commands, as our sages taught that the sons of Noah were commanded with seven commandments: laws, blasphemy, idolatry, illicit sexual relations, murder, theft, and eating tearing a limb, and we say [in the Gemara], how do we know this... and R. Chananyah b. Gamliel adds [the prohibition of eating] blood from a live animal... [and other sages add other commandments] and these continue to be added upon until the number of commandments "of hearing" [meaning, explicitly directed by God] amounts to twenty-eight, and some say thirty
The number thirty is also mentioned in the Talmud, Hullin 92a (also in the Talmud Yerushalmi, Avodah Zarah 2:1). Rashi there says that he does not know what these are precisely, but in fact another very early commentator, R. Shmuel ben Hofni Gaon (Baghdad, c.950-1035) in his commentary to Bereshit 34:12, gives us a list of those thirty commandments:
A very similar list of 30 commandments is compiled by R. Azariah of Fano, in his book Asarah Maamarot, although he doesn't count prayer, sacrifices, and God's unity; instead he counts an obligation to procreate, a prohibition against crossbreeding plants, stirking a Jew, and the obligation to honor and study Torah without studying the parts of the Torah that are the exclusive heritage of the Jewish people.
Rabbinic sources clearly assume that whatever was commanded of Noah's descended are obligatory, even on pain of death. However, it is not clear if all of these commandments, especially beyond the seven primary commands, are obligatory to the same extent, or merely recommended, or something in between. Rambam, for example, mentions (Hilkhot Melakhim 10:8) that non-Jews are prohibited from crossbreeding animals, but not punished by a Jewish court for violating this command.
The most complicated aspect of your question is the relationship between the seven/thirty commandments and what might be obligatory "on the basis of morality and ethics". The earliest rabbinic commentators believed that the seven/thirty commandments are independent from reasonable ethics. In other words, moral laws have a self-evident binding force all on their own, even if God had not said anything about them. This is pointed out by the commentators on the earliest stories of the Torah: Kayin is held morally responsible for murdering his brother, the wicked in the time of Noah were guilty for their sins, etc. Thus, R. Saadiah Gaon (10th century) writes (commentary to Bereshit 4:3) that he is unsure precisely how many commandments the earlier pre-Sinai generations received from God, but they must have gotten some instruction from Him, because otherwise Kayin and Hevel would not have thought up the idea of sacrifices on their own (since nobody today would consider animal sacrifices to God as self-evidently moral). R. Nissim is even more explicit about this in the source I quoted above, where he writes:
כל המצוות שהן תלויות בסברא ובאובנתא דליבא כבר הכל מתחייבים בהם מיום אשר ברא אלוקים אדם על הארץ עליו ועל זרעו אחריו לדורי דורים והמצות שהן נודעות מדרך השמועה מדברי הנביאים לא חשך אלהינו מלחייב לקדמונים מה שהיה ראוי בעין חכמתו לחייב לקדמונים מה שיה ראוי בעין חכמתו לחייבם. ... אינן כולן של שמועה כי חיוב ידיעת מציאות הקב"ה ולשמוע בקולו ולעבדו מדין הדעת הן ראויין, ושפיכות דם נקי והגזל מדרך השכל הן אסורין, ולא נצטרפו מצות השמועה שניתנו לאחר מיכן אלא עם אותן שהן של שמועה כיוצא בהן
All "commandments" which are dependent upon reason and human understanding are incumbent upon everyone, from the day that God created man on Earth, upon him and upon his progeny after him for all generations. And the commands which are known through 'communication' from the words of the prophet are not withheld by God from being obligatory upon the ancients whatever was deemed fit by His Wisdom to obligate them... although not all of them are 'of communication' [that would be unreasonable if not directed by God], for the obligation to know of God's existence, heeding His voice, and serving Him are all fit [to be obligatory] by force of intellect, and the murder of innocents or theft are prohibited by way of reason
In other words, yes it is true that some of these commandments overlap with what human reasoning might come up with as morally obligatory, but some cannot be identified with human reasoning and must have been commanded by God directly (as R. Saadiah Gaon stated), and so at the end of the day, all of these listed obligations are sourced in God's explicit command. Likewise, Rambam introduces the seven laws of Noah by saying אף על פי שכולן קבלה הן בידינו ממשה רבינו, והדעת נוטה להן, מכלל דברי התורה ייראה שעל אלו נצטוו - although these are all received traditions from Moshe, and the mind is inclined towards them, from the general words of the Torah it can be seen that these were commanded [and not derived from human reason].
There is another school of thought which does appear to identify the "seven laws" with a morality that could have been derived by human reason. Such an idea might be found in "Midrash Lekah Tov" by R. Toviah b. Eliezer (11-12th century, Salonica) in his commentary to Bereshit 2:15 who attempts to prove that reference is made to all these laws (although it is not clear to me if he believes that Adam was, in fact, commanded about them specifically or expected to have reasoned his way to them):
כי כל אלה הן מצות הדעת שאפילו לא נתנה תורה לישראל נוהגין היו בהן הדורות לשמרן בדעתן וכן הוא אומר באברהם כי ידעתיו למען אשר יצוה את בניו ואת ביתו אחריו ושמרו דרך ה' לעשות צדקה ומשפט כי אנשי דור המבול מפני החמס נתחתם עליהם גזר דין שלא היו נוהגין על המשפטים. ברכת השם עברו דור הפלגה ולקו שנאמר ונעשה לנו שם שכפרו בעיקר. על זה היה זריז בו איוב שנאמר אם אראה אור כי יהל וירח יקר הולך ויפת בסתר לבי ותשק ידי לפי גם הוא עון פלילי כי כחשתי לאל ממעל' שלא עבד לא לשמש ולא לירח מכלל שהיו אחרים עובדים שפיכת דמים נענש בו קין גלוי עריות נענש בו אבימלך דכתיב השב אשת האיש ואומר ואיך אעשה הרעה הגדולה הזאת וחטאתי לאלהים גזל הוא החמס אבר מן החי שנאמר אך בשר בנפשו דמו לא תאכלו וכל אלה שהוזכרנו עונשן לא ענש אלא אם כן הזהיר למדנו כי נצטווה אדם הראשון בכל אלה
More explicitly, Ramban (13th century) writes (Torat Hashem Temimah, p. 173 in Kitvei ha-Ramban, ed. Chavel)
על דרך הפשט אלו מצות שכליות הן וכל נברא מכיר את בוראו צריך להזהר מהן According to the straightforward understanding [as opposed to the more interpretative reading of the Sages] these are 'reasonable commandments,' and all creatures who recognize their Creator must be careful about them
There is much more to discuss here and many more sources to bring, but we have other things to clarify.
Regardless of whether you view the 'seven commands of Noah' as being fully identifiable with moral reasoning, or as being obligatory only because of God's explicit command, or something in-between, nearly all the Jewish thinkers agree that every person, whether Jew or non-Jew, is also obligated to behave ethically, even if such behavior does not fit exactly into any of the stated "commandments". Almost nothing is Jewish thought can really be said to be universally accepted, with no dissenters, but my impression is that this is the position of the overwhelming majority of Jewish thinkers. Just to quote Rambam (Dalalat al-hairin/Moreh Nevukhim, 3:17):
God must always be considered just, in that He rewards the righteous on all his deeds from his acts of kindness and rightness even though he was not commanded as such by a prophet, and He punishes all evil acts that a person does even though he was not warned as such by a prophet, for he is warned as such by way of nature, meaning to say, the warning against oppression and injustice
[it happens to be that this quote is in the context of Rambam quoting an opinion he disagrees with somewhat, but he clearly does agree with this aspect].
We have thus far considered two sources of obligations for all of humanity: explicit commands (numbering either seven, thirty, or something in between), and moral reasoning. However, there is another possible source of obligation which you've alluded to in your question: derivations from those original divinely instructed commands. There are many rabbinic sources attesting to the fact that each of the seven laws (if that is indeed the right number) actually include what in the Jewish code list of 613 are numbered as several commands.
For example, "illicit sexual relations" is counted once in the seven listed by the Talmud in Sanhedrin and by Rambam, but it is clear that this actually includes several prohibitions: male same-sex relations, sibling incest, adultery, bestiality, etc. as Rambam details (in Hilkhot Melakhim 9:4). Instead of listing them out, I would recommend reading that chapter in Rambam's Hilkhot Melakhim, which is readily available online. Another medieval source which takes this view and even adds to Rambam's list is the Sefer ha-Chinuch (possibly written by R. Pinhas haLevi of Lunel, 13th century). He adds the following (parenthesis refer to the mitzvah number, out of 613, where he writes this):
In addition to these medieval writers, some modern books have been written in the past few decades that are centered around this idea and provide more details on the derivations of these seven laws. One such book is "The Seven Laws of Noah" by Aaron Lichtenstein (not this man, another Aaron Lichtenstein)
Everything discussed so far has assumed that the non-Jew is obligated in these listed commands. However, one could argue that the Torah's guidelines for the lives of Jewish people hints to the fact that other actions and behaviors might be valuable even for those who are not commanded to do so. In other words, if you assume (as many traditional Jewish thinkers do), that the Torah's laws provide blessings and are a guide to the good life in this world and/or the afterlife, then it stands to reason that a non-Jew who wants a share in those blessings would be encouraged to try following the same rules. This answer has already gotten too long, but basically, there are lots of different opinions here, and it depends on how you view the Torah's commands more generally. I have already mentioned Rambam several times, and so I'll point out that his opinion appears to be that a non-Jew can indeed pick up any mitzvah for his own spiritual growth, as long as he is doing so with the intent of following God's Torah, and not doing so in order to invent a new religion (this is clear from his commentary to the Mishnah Terumot 3:9, and a letter (no. 60) on the topic. However, one of the greatest halakhic authorities of the 20th century, Rabbi Moshe Feinstein, refused to believe that Rambam held such a view, and stated emphatically that there is no value whatsoever in a non-Jew's performance of ritual commands (Iggerot Moshe Y.D. 2:6)
The Mishna Berurah (Biur Halachah 304) writes that a ger toshav can obligate himself in any of the 613 mitzvos, except for kashrus. According to this position, if a gentile upon his decision to become a ger toshav would choose to add to the required 7 Noahide Obligations, they would become obligatory for him.
אחר כתבי כ"ז התבוננתי שאפשר לקיים דבריו דס"ל להמגן אברהם ג"כ דאינו בר ישראל כלל ובכלל גר תושב הוא והכל כאשר כתבנו אלא דס"ל דגר תושב גופא אם רצה לקבל עליו בעת תחלת גירותו עוד מצות מלבד השבע ג"כ חלה קבלתו שמחוייב אח"כ לקיים אלא דמה דנקטו שבע רבותא אשמועינן דאף ששבע מצות מחוייב לקיים כל בן נח ומאי רבותייהו אפ"ה חלה הקבלה ובכלל גר תושב הוא לענין שמצווין להחיותו וכ"ש אם קבל עליו יתר מצות בודאי מהני ולא תקשה ע"ז דאיך ישמור שבת והלא גר תושב ג"כ אסור לשמור שבת כדמוכח ביבמות מ"ח ע"ב בתוד"ה זה גר וכו' דזהו בסתם גר תושב שלא קבל עליו רק שבע מצות כנהוג וא"כ הוא לענין שאר מצות כא"י גמור משא"כ כשקבל עליו עוד מצות בתחלת גירותו ובכללם היה ג"כ שבת בודאי יכול לקיימם ומחוייב לקיימם ומה דאיתא בבכורות דא"י שרצה לקבל כל התורה חוץ מד"א אין מקבלין אותו היינו לענין לעשותו ישראל גמור אבל לא לענין גר תושב
Besides the 7 mitzvos of noach. Many other obligations they have seem to be in dispute. Rabeinu nissim in hakdama wrote that goyim are obligated to keep mitzvos sichlios (logical commands). The chizkuni noach 7:21 says the same thing.
(Even though many of the 7 mitzvos are also logical to keep, the reason why the 7 mitzvos are different is because they're obligated to death by beis din for violating them)
Yechezkel 16:49 implies that Sedom got punished for not giving tzedeka. Indeed the Ran (Sanhedrin 56b ויצו) writes non jews are commanded to give tzedaka. However Rav Elchanan Wasserman (kovetz parshios וירא)deduces from the Rambam that they have no such obligation, since the Rambam doesn't mention it. Rav Elchanan himself gives a few suggestions why sdom was punished for not giving tzedaka. First, he says according to Rav Chaim Vital that the Torah doesn't have explicit commands on midos is because it's a fundamental to have to keep the Torah. He says therefore non jews as well are obligated to have good middos, because if not they wouldn't be able to keep their 7 laws. Therefore Sdom got punished for being cruel and not giving tzedaka. (This seems to be similar to those that hold goyim are obligated in logical mitzvos) He also suggests that even though sdom wasn't obligated to give tzedaka, but if they did the merit would've protected them from being punished. Some (Rav Nechemia Kaplan shlita) bring a proof that non jews are obligated in logical mitzvos from bireishis rabba p.36:7 that says hashem exempted Avraham from the mitzva of cibud av. Why would he need an exemption if he wasn't obligated to keep the Torah? Must be since its a logical mitzva. However the gemara in kiddushin 31a brings a story with a non jew who gave up lots of money to honor his father and got rewarded. The gemara was showing how much reward a person whose not commanded in something gets. Yet if non jews are obligated in logical mitzvos how come the gemara says he wasn't commanded? Avnei Nezer on nazir suggests that since mitzvos sichlios aren't explicit in the torah one gets less reward for it like one whose not commanded. Rav nechemia Kaplan suggested that specifically in that case it wasn't logical to give up so much money for honoring his father. (Also perhaps you can suggest according to gemara baba kama 38a that non jews lost the reward on mitzvos as if they were commanded. They only get reward as if they're not commanded)
Reb Moshe (Igros YD 2:7) seems to argue and hold they're not obligated to do logical commands but if they do them they get reward. However if they keep 613 mitzvos which aren't logical to keep they get no reward for them at all. The Brisker Rov (likutei hagrach siman 56) also holds non jews don't get reward for keeping 613 mitzvos. However Rav Berel Povarsky (Bad Kodesh sukkos) holds non jews do get reward for keeping 613 mitzvos (except for sukka, as he explains there). Part of the dispute is how to learn up the Rambam 10 melachim 9-10. The radvaz on the Rambam learns that non jews do get reward for keeping mitzvos howevet they shouldn't do mitzvos that have holiness connected to them such as teffilin. ( Im not sure if he's saying they get no reward for it, or they shouldn't do it for other reasons) Rav Shlomo Zalman Aurbach ztl (SA 17 Dirshu Mb) seems to hold goyim get rewarded as well. Since he says a goy that puts on tzitzis gets reward, however he cant make a blessing on it, since he's not obligated.
The Rambam (10 melachim 6) poskins non jews cant interbreed animals and trees (in certain ways), however hes not killed if he does so. You should just see the rambam melachim perek 10 to see a bigger description of what non jews can or cant do, which is more than 7 mitzvos. A non jew that studies Torah is obligated to death, but not killed.(Rambam 10 melachim 9) However if he studies the 7 mitzvos he's as great as a cohen gadol (Baba Kama 38a). The last Drisha 48 brings from his master that just like women are obligated to learn their laws and should make a birkas hatorah, so too non jews are obligated to learn their laws and make a birkas HaTorah.
The shieltos by mitzvas pru urvu holds non jews are obligated in the mitzva of pru urvu. However Tosafos Yevamos 62a says based off the gemara sanhedrin 59b that non jews aren't obligated in pru urvu. (The mefarshim on sheiltos try to answer up that gemara)
If a non jew accepts 7 mitzvos and becomes a ger toshav, there's a duspute amongst the reishonim if hes obligated to keep shabbos.
The answer is yes, there are more the 7 mitzvos and recommendations for non jews to keep. Although many things are disputed. However making a full list will take a very long time.
This is a more complex question than might be thought. The short answer, in the context of your precise wording would be, no, gentiles should not follow more than the seven Noahide Laws.
But that "no" is said with qualifiers. The seven commandments of Noach are said in a similar context to the idea that there are 613 commandments for the Jewish people in the Torah. That number is applied to what are described as the roots of the commandments.
So just as with the 613 root commandments, there are countless laws, also referred to as mitzvot (which has a connotation of connections), that derive from those roots. The same is true with the seven commandments of Noach.
Per your request, in regard to a very comprehensive, list form together with a detailed explanation of these derivatives, you should look to the book written by Rabbi Moshe Weiner on the subject of the Seven Commandments of Noach.
In the Hebrew original it is titled, ספר שבע מצות השם.
The latest edition translated into English is called, The Divine Code.
Here is a link to a video describing The Divine Code.
For gentiles interested in learning more about how the Bible traditionally connects to all people on earth, this text comes highly recommended.
In that context, it should be noted that the sum of all G-d's commandments found in the Torah, meaning the 613 commandments directed toward the Jewish people together with the 7 commandments of Noach, (which are directed to the balance and complement of the whole of mankind,) totals 620, which in Hebrew letters is כתר, which means Crown. That through all of us working together in service to the Creator of us all, we reveal G-d's Crown of Kingship over all of creation. This is the deeper understanding of the closing words and law to Rambam's Mishneh Torah, Laws of Kings, chapter 12:5 which discusses the final redemption and the coming of Moshiach which says:
In that era, there will be neither famine or war, envy or competition for good will flow in abundance and all the delights will be freely available as dust. The occupation of the entire world will be solely to know God. Therefore, the Jews will be great sages and know the hidden matters, grasping the knowledge of their Creator according to the full extent of human potential, as Isaiah 11:9 states: 'The world will be filled with the knowledge of God as the waters cover the ocean bed."
Meiri, Sanhedrin 59a, states that a gentile can perform other Mitzvot for the sake of heaven (resting on Shabbat or Yom Tov is an exception):
ומ"מ כל שהוא עוסק בעיקרי שבע מצות ובפרטיהם ובמה שיוצא מהם אע"פ שרוב גופי תורה נכללים בהם מכבדין אותו אפילו ככהן גדול שאין כאן חשש לטעות אחריו שהרי אף בשלו הוא עוסק וכל שכן אם חקירתו על דעת לבא עד תכלית שלימות תורתינו עד שאם ימצאנה שלימה יחזור ויתגייר וכל שכן אם עוסק ומקיים עיקרי מצותיה לשמה אף בשאר חלקים שבה שלא משבע מצות:
