rosends got it right.
But I'll go through the quotes anyhow.
A. Non-Jews aren't humans. Keritot 6b.
There are a handful of specific technical laws in the Bible that pertain to "an adam" which the Talmud interprets as "Jews only"; for a non-Jew we are more lenient. The idea simply is that most of the Torah's laws were intended for a Jewish audience, so ...
These attacks are usually amalgamations of the following:
Pure invention -- some of the books listed don't exist or the quotes are fabrications
Mistranslations or selective quoting
Out of context quotes (statements made in the course of a protracted legal argument presented as definitive statements of belief or statements made to make a legal point being ...
R' Nachman's actual statement from Likutei Moharan II 48 is:
וְדַע, שֶׁהָאָדָם צָרִיך לַעֲבר עַל גֶּשֶׁר צַר מְאד מְאד וְהַכְּלָל וְהָעִקָּר שֶׁלּא יִתְפַּחֵד כְּלָל
"And know, a person needs to make his passage on a very, very narrow bridge, and the rule and the essence is to not be afraid at all." (*)
Likutei Moharan II 24:
One of the sources for this statement is in Ma'amar 2, towards the end of Chapter 30 of
Rabbi Yosef Albo's Sefer Ha'ikrim. It is brought in the name of the "chacham" - "wise person."
אי אפשר שישיג עצמותו שום נמצא זולתו. כמו שאמר החכם כששאלו אותו אם היה יודע מהות האל, והשיב: אילו ידעתיו הייתיו. כלומר כי אין מי שישיג עצמותו אלא הוא יתברך, עם היות מציאותו ...
אוקימתא (from להקים - to put up) is putting a statement in a specific more elaborate way.
In short, when there are two (or more) contradicting sources that bring seemingly opposite opinions, Ukimta is used to limit each statement's application to a specific, non-overlapping situation that eliminates the contradiction. For example, Brochos 45b:
תני חדא ...
Rabbi Re'uven Brauner wrote a pamphlet indexing verses used in prayer called "Shimush Pesukim" (in halakhah.com): http://halakhah.com/rst/pesukim.pdf
However, the specific verse you quoted is not found in that index (nor do I remember it), except for a haftarah.
רמ"א, תורת העולה, ח"א פרק ו
The Rema in תורת העולה, ח"א פרק ו says
וכבר כתב חכם אחד שלא טעם טעם שמחה מי שלא טעם התרת הספיקות השכליות
and a wise man said that someone who has not tasted the resolution of
(intellectual?) doubt has not tasted joy,
in connection with שמחת בית השואבה !
Metzudat David is from “18th century David Altschuler” and the
Nesivei Yeshurun al HaTorah1 (פרשת תולדות) cites a story where the Sefas Emes implies that he started at a lofty spiritual level due to the greatness of his illustrious grandfather, the Chidushei HaRI"M. To convey this idea, the Sefas Emes employs the analogy of a small child born at the summit of a lofty mountain:
ואלה תולדות יצחק בן אברהם אברהם הוליד את ...
Apparently it is quoted by one of the Chasam Sofer's students, Rabbi Eliezer Lipman Naizatts, in his sefer Mei Menuchot, page 43a, as having been said by the Chasam Sofer in front of many of his students.
Follow the link:
Most likely, you're thinking of Yehuda ben Teima in Avot 5:21, which is of the form you describe, though with a different particular list:
הוּא הָיָה אוֹמֵר, בֶּן חָמֵשׁ שָׁנִים לַמִּקְרָא, בֶּן עֶשֶׂר לַמִּשְׁנָה, בֶּן שְׁלשׁ עֶשְׂרֵה לַמִּצְוֹת, בֶּן חֲמֵשׁ עֶשְׂרֵה לַתַּלְמוּד, בֶּן שְׁמֹנֶה עֶשְׂרֵה לַחֻפָּה, בֶּן עֶשְׂרִים לִרְדֹּף, בֶּן שְׁלשִׁים ...
In Bava Kama 92b, Rava asks Rabba bar Mari for the source of a popular saying along the same lines. Rabba bar Mari answers with five different sources for it. The third verse, which is attributed to "the Writings," is in fact the same verse from Ben Sira that you suggested as the source.
א"ל רבא לרבה בר מרי מנא הא מילתא דאמרי אינשי מטייל ואזיל דיקלא ...
The Torah writes about Moshe that he remained youthful and vigorous until 120 years. We thus bless each other with the designation "Until 120" with the same connotation, viz. that they should live a long life without any physical, emotional and intellectual degradation.
וּמֹשֶׁה, בֶּן-מֵאָה וְעֶשְׂרִים שָׁנָה--בְּמֹתוֹ; לֹא-כָהֲת
This prayer is Jewish. In the 11th-century, Solomon ibn Gabirol, the author of Adon Olam, wrote [Mibchar ha-Peninim (Choice of Pearls), Chapter 17 (Consciousness), verse 2]
At the head of all understanding is realizing what is and what cannot
be, and consoling ourselves for what is not in our power to change.
I also used to think it had to do with Moshe's age. But then this made me wonder why we say "You should live till 120", since it sounds more like a curse than a blessing to limit someone's potential age to a set number of years. Then it was pointed out to me that it isn't a curse, but a determination made by HaShem, explicitly stated in the Torah.
There are certain cases in which it is appropriate to kill someone, and certain cased in which killing is inappropriate. Typically appropriate killing would be legal executions. For discussion of when extrajudicial killing is justified see here. The quote doesn't say anything about which cases are which. Rather, it says that killing certain people is ...
From Masechet Bava Batra, 158b
בחזקת מי ר' אילא אמר בחזקת יורשי האם ר' זירא אמר בחזקת יורשי הבן כי סליק רבי זירא קם בשיטתיה דרבי אילא קם רבה בשיטתיה דרבי זירא אמר רבי זירא שמע מינה אוירא דארץ ישראל מחכים
The mishna states that according to Rabbi Akiva, the property retains its previous ownership status. The Gemara asks: In whose possession does the ...
At the K'nesia Gedolah in 1980, the "Lev Simcha" - the Rebbe of Gur - quoted this statement in the name of Rav Shapira from the first Siyum HaShas.
[The Rebbe started Daf HaYomi of Yerushalmi at that time, and added that the words דף של ספינה - a plank from a ship - have the same numeric value as זהו ירושלמי - this is Yerushalmi.]
Regarding the first quote "The whole world is a very narrow bridge..." I can only find "A man must traverse a very narrow bridge...". Perhaps the songwriter wrote the song based on those words which are attributed to Rabbi Nachman.
Regarding "It is a great mitzvah to be happy constantly". This is mentioned in Likuteh Morahan Tinyana 24
מצוה גדולה להיות ...
I contacted someone who asked the rabbi from whom I'd heard it. It was the Chazon Ish.
(Which makes sense; he wrote a lot about free will. Fascinatingly, it was his opinion that when we pray for there to be less evil in the world, we are praying that G-d tamper with wicked people's free will. R' Moshe Feinstein disagreed.)
I'm not sure to which Rav you're referring, but the Gemara in Masechet Shabbat 104A (Hebrew, English) states that "one who comes to defile himself is given an opening (i.e. he is permitted, but not actively helped) and one who comes to cleanse himself [...] is helped":
בא ליטמא פותחין לו בא ליטהר מסייעים אותו
"The Rock, His work is perfect; for all His ways are justice; a God of faithfulness and without iniquity, righteous and just is He. Is corruption His? No; His children's is the blemish" - Deuteronomy 32:4
The answer seems to be most probably R. Chaim Soloveitchik. This is recorded here. Additionally, Ishim V'shitot of R. Zevin (ed. Kol M'vaser, 2007, pp. 51-2) attributes it to him:
מחמיר לפי הגדרתו של ר' חיים עצמו. וכך היה אומר: כלום אני מקיל באיסורים? אדרבה אני מחמיר בפיקוח נפש.
This is consistent with R. Soloveitchik's approach to such matters, as R. ...
To add to Avrohom Yitzchok's answer:
The B'nei Yissaschar (Sivan 5:13; cf. Kisleiv 3:19) attributes this statement to the Rambam:
ואמר הרמב"ם מי שלא טעם טעם התרת הספיקות (בתורה) לא טעם שמחה מימיו
And the Rambam said, "Whoever has not tasted the taste of resolving doubts (in Torah), has not tasted joy all his days."
I don't know where in ...
One place this is mentioned is the Chafetz Chaim's Toras HaBayis (ch. 10). The Chafetz Chaim notes the symbolism in how the angels on the ladder in Ya'akov's dream were either ascending or descending but could not remain static on the same level (B'reishis 28:12).
The Chafetz Chaim generically cites "the holy sefarim" for this interpretation of the verse, ...
Interestingly, although widely quoted, a cursory search shows me only 16th century sources for this exact wording, although they quote "Chazal".
E.g. R. Isaiah Horowitz in Shelah to Parshat Vayishlach:
כי אם על יוצאי חלציו גם כן, כי מעשה אבות סימן לבנים
"But to his descendants as well, for the act of the ancestors is an indicator for the descendants."