Is every Jew who is capable of doing so, required to learn "kol ha'torah kula"? If yes, what would that include?
"You are not responsible to finish the work (Torah learning), yet you are not allowed to quit (learning) either." - R' Tarfon, Pirkei Avos 2:16
Since the Torah is the will and wisdom of Hashem, it has no end. So, you are not obligated to "finish" it. You must simply keep learning it. You shall toil in Torah learning day and night constantly. (See The Book of Joshuah 1:8)
So, by definition, Torah includes an infinite amount of wisdom, even though we have known categories to identify its contents, like: Written, Oral, Tanach, Shas, Medrash, Halachah, Hashkafa, Kabbalah, etc. You must simply accomplish as much as you can as often as you can.
The actual Mishnah quoted does not specifically say the "work" is "learning Torah. However, it can be readily seen from context that it is what R' Tarfon means. (There are many mefarshim that explain R' Tarfon's words this way; as specifically referring to the mitzvah of learning Torah.)
The very next words of R' Tarfon in the Mishnah say "..If you have learned much Torah...". This context tells us that the whole Mishnah refers to the mitzvah of learning Torah.
The Rambam in his Peirush HaMishnayos on Avos , The Tosfos Yom Tov, Rav Ovadiah MiBartenura, and the Meleches Shlomo, as well as the Maharal of Prague all interpret Rabbi Tarfon as describing the mitzvah of learning Torah.
The Rambam in the Yad HaChazakah, Laws of Talmud Torah (in Sefer Mada) 3:6 says about the mitzvah of learning Torah: "..you are not responsible to finish the work, but you may not stop from it...". This exactly echos R' Tarfon's wording.
A source in Gemara for this idea can be found in Menachos 99b; a few lines from the bottom of the page.
"The House of R' Yishmael taught: The words of Torah should not be upon you like an obligation, but you also may not exempt yourself from them either."
Rashi explains: "..Like a man who has a debt and says "Oh when will I be done with this debt?" So a man must not say: "I will learn one chapter and then have fulfilled the obligation." for you have no permission to exempt yourself from them (the words of Torah learning)."
Tosfos in his final explanation of the above words of R' Yishmael's House:
"You do not have to learn the whole Torah. - as it is taught already in the Mishnah (Avos 2:16) "You are not responsible to finish the work, but you may not quit either."
I hope this helps. :)
In codifying the obligation to study Torah, Rambam never delineates any set components of Torah that a person must study (in the first chapter of Hilchot Talmud Torah, in the rest of that section, or anywhere else that I am aware of).
Accordingly, we can infer that there is no obligation to learn "kol haTorah kulah".
This inference is made by Heshy Zelcer as well:
Studying the entire Torah: Is there a requirement to study the entire Torah including the Written and Oral Law? Rambam clearly expects one to learn all of the Written and Oral Law early on in his learning process, but he does not seem to formalize this obligation.
Source: Ḥakirah (vol. 2 p. 125)
However, presumably Rambam would agree that one must learn practical halacha (something he places great emphasis on in numerous places, e.g Perush HaMishnayot to Niddah (6:15)), for otherwise, how would one know how to practice halacha.
The Semag, however, implies that besides for the obligation to learn practical halacha, there is an obligation to study each of the 613 mitzvot. In his introduction to the Positive Commandments in Semag, he writes:
כבר הקדמתי בספר מצות לא תעשה עניין ספר הזה כי הוא לידע יסוד המצות על פי הקבלה ולא [כל] חילוקיהם באורך ויש מצות שאין נוהגים לדורות ולא היו כי אם לפי שעה ואין ראוי למנותם כגון ברכות וקללות דהר גריזים והר עיבל וסיד אבנים גדולות ועשיית המשכן וכיוצא באלו:
ויש מהמון [ה]עם שאומרין מה לנו ולמצות סדר קדשים קל וחומר למצות סדר זרעים ולמצות סדר טהרות לדברים שאין נוהגין בזמן הזה, אל יאמר אדם כן כי המצות אשר צוה אדון העולם יש לידע יסודותיהם אף על פי שאינם צריכין עתה כי על כל המצות נצטוינו ולמדתם אותם (דברים יא, יט) ונאמר (דברים כז, א) שמור את כל המצוה אשר אנכי מצוה אתכם וגומר, ונאמר (ויקרא כב, לא) ושמרתם מצותי ועשיתם אותם...וכן אמרו רבותינו (תו"כ אמור פרק ט ה"ג) ושמרתם זו המשנה ועשיתם זו העשייה. ויש אדם שילמוד כל ימיו ולא יכול להשיג לדעת מצוה אחת כמאמרה בעל פה מחמת אורך הגמרא ופלפולה...
That is, there is an obligation to know the basics of all of the mitzvot - even the ones which may not apply to the learner.
It should be noted, that like Rambam, I am unaware of Rif, Rosh, Tur, the Shulhan Arukh, (or any other Rishon, for that matter) expressing the Semag's view; or delineating a particular corpus that a person is obligated to master.
Additionally, neither Rambam (Hilkhot Talmud Torah 1:1) nor Semag (Assei 12) obligate women in the study of Torah.
If you are interested in the term "kol haTorah kulah" note that in Hazal it often (always?) means the entire Pentateuch. See for example, the Sifrei to Devarim (Piska 1) that Moshe Rabbenu wrote "kol haTorah kulah"; i.e. the Pentateuch.