Is there a requirement to have a meal on the night of Meggilah reading?
If not, why is there a custom to eat one?
Is one required to drink even at night?
The Talmud (Megillah 7b) very clearly rules that one who ate his Purim meal at night has not fulfilled his obligation to have a meal on Purim.
The Mordechai (Moed, Remez 787) quotes the Raavyah who (as understood by the Bach OC 695) rules that the night of Purim should have a seudah, and the Gemara is only saying that the obligation for the main meal must be during the day, parallel to Shabbat where one must eat a meal at night and during the day, but the day meal is considered more significant (see Pesachim 105a).
Interestingly, the Beit Yosef (OC 695) quotes Rav Hai Gaon that one who took an oath not to eat on Purim day, should eat his obligatory seudah on Purim night, and not break his oath, implying that there is value, at least post facto, to a seudah at night.
Just to present the other side, the Kol Bo (Siman 45) mentions a custom not to eat meat on Purim night lest one think one has fulfilled one's obligation. However, the Elya Zuta (695:3) already notes that this custom is no longer followed, but rather festive meals are held per the Bach/Raavyah/Mordechai quoted above.
It is a Mitzvah to be joyous and have a small Seudah on Purim evening as well. (Ram”a 695:1)
The Sfas Emes implies that there is no inyan of drinking at night (Likkutim to Meggilas Ester 9:17).
Although a responsum of the Tshuva Meiahava (vol. 2: Orach Chaim 299) has been quoted as a source for drinking at night, what he acy
ורבים אומרים נשקנו יין גם הלילה... לא מפני שאוהבים את מרדכי, אלא לב כסילי' בבית השמחה
The context there is that he is writing the tshuva whilst surrounded by drunken morons and his pithy wordplay does not present them in a better light. This is certainly not a source for drinking at night and if anything implies the opposite.