1. Does it fulfill a mitzvah?
Not only can it fulfill one mitzvah, but three:
The Malbim (Shemos 20:2), Maharam Schik (Mitzvah 26) and S
1. Does it fulfill a mitzvah?
Not only can it fulfill one mitzvah, but three:
The Malbim (Shemos 20:2), Maharam Schik (Mitzvah 26) and Seforno (beginning of Ohr Amim) all explain that the commandment of אנכי ה' אלוקיך, believing in God, is a command to philosophically justify those beliefs
The Chovos Halevavos (introduction) explains that there is a separate command to "know God", as evidenced by the verses ידעת היום והשבות אל לבבך כי ה' הוא האלקים (Devarim 4:39). He also connects this to the second mitzvah (as counted by the Rambam) of belief in God's Oneness
The Ramban in Shaar HaGemul doesn't discuss the nature/existence of God, but he does discuss questions of how God runs the world. He writes "חובת כל הנברא עובד מאהבה ומיראה, לתור בדעתו לצדק המשפט ולאמת הדין כפי מה שידו משגת", there's an obligation upon those who serve God to justify His actions.
Even without fulfillment of a specific mitzvah, there's still value in asking questions or 'philosophizing' about the truths of Judaism, because it fosters a deeper connection with Judaism (see Divrei Negidim Hagadah by the 4 sons, and R. Hirsch's Nineteen Letters, no. 18)
2. Can Philosophy Teach Something that is not in the Torah?
The whole point of utilizing philosophy is that, in some areas, the Torah might not be sufficient. This does not necessarily mean that these truths aren't contained in the Torah, but that only after understanding certain principles (which are better taught from external sources) can one understand the Torah properly. Thus, the Rambam writes in several places (see M.N. 1:71 and 2:12, and Hilchos Kiddush Hachodesh 17:24) that wisdom which he learned from philosophers and scientists originally belinged to the Jews, but was lost due to the exile.
3. Does any "Rishon" base their beliefs fundamentally/entirely upon philosophy?
If by "rishon", you mean traditional Torah scholars who have always been totally part of the Jewish religious community, than the answer is no. There was, for a period of time (during the Rishonim) a school of Jewish rationalists who pretty much believed entirely in philosophy and only tried to fit the Torah in to their philosophical worldview, but these were rejected by the mainstream community. An example of such a person is (Rabbi) Levi ben Avraham, who was excommunicated by the Rashba for his extreme views, and others of his ilk. (With perhaps one very early exception)
The real question is one of extent. Some rishonim thought that we should accept much of what would be taught by philosophy, even if it may be a bit of a stretch in terms of how it fits with maamarei chazal, others thought that we should ignore philosophy completely, and of course there are many in between. Despite this, while the Rambam may have used philosophy to an extent that other rishonim were uncomfortable with, he still writes that one must subjugate their knowledge to the Torah, and not the other way around (end of Hilchos Me'ilah)
4. Did any Rishon prioritize philosophical understandings over simple reading of the Torah?
Yes, but not because they thought philosophy to be above the Torah, ח"ו, but because they felt that if something was proven philosophically, then it must be that the Torah means something other than its literal meaning. There are many specific examples of this, but as a methodology, is specifically mentioned by the Rambam (M.N. 2:25) and R' Saadia Gaon (Emunos V'Deos 7:1)