What sources are there for our minhag of doing a Bar Mitzva Seuda (meal), and do the sources state that the seuda must specifically be done on the day of the Bar Mitzva (or the night of)? And is this seuda actually a Seudas Mitzva?
See the Magen Avraham, Orach Chaim 225:4.
He points out a couple things:
It is a Mitzvah for the father to make a Seudah (festive meal) on the day his son turns 13 (enters his 14th year, i.e. not the evening he turns 13, but the next day). This is equal to the obligation to make a Wedding Seudah.
If the son makes a Drashah (Torah speech/lecture) in honor of his Bar Mitzvah, this is considered a celebration of the Bar Mitzvah (and the meal is therefore a Seudat Mitvah) even if it is not on the actual day.
The Magen Avraham quotes his sources, but I did not look them up.
Cf. the following passage in Zohar Chadash, Parshas Bereishis: "R. Shimon Bar Yochai invited the leading scholars of the Mishnah to partake of a great festive meal which he had arranged.... He was very happy..., because, [as he explained,] 'On this day a holy and exalted soul descended...into my son R. Elazar, and on this joyous occasion I shall experience supreme joy.' "