Very thorough essay question, but interestingly I would have to disagree with it, which would make for an interesting response! Firstly, I do think this is a successful comedy - there are clear moments of hilarity using all the usual Shakespearian methods of mistaken identities, cross dressing, characters falling in love with one another and human folly exhibited. However, my issue is that it is clear in this play that all the conflicts are not resolved, and there are certain issues that serve to question the play's status as a comedy - to put it bluntly there are parts that some could argue are just not funny.
This is clearly seen in the character of Malvolio and the revenge that is taken against him by Sir Toby Belch, Maria and Feste. Note how, whilst Sir Toby and Maria marry at the end, they flee Olivia's household, afraid of what will happen to them. They are excluded from the "happy ending" just as much as Malvolio is. In Act V scene i he presents the one discordant note as he is finally released from his prison and, after having the elaborate ruse exposed, storms off from the merry couples with the line:
I'll be revenged on the whole pack of you!
We get the feeling that whilst a certain amount of judiciously administered punishment is correct to settle the kill-joy character that Malvolio is and to prick his self-importance, after Olivia confines him into the care of Sir Toby, things deteriorate rapidly, and his punishment does definitely not fit the crime.
For another character who is left outside of the happiness at the end, consider Antonio, who is clearly in love with Sebastian, yet sees him marry a woman whom he hardly knows and abandon him. Although this play is definitely a comedy, it appears that there are a number of characters who do not share the "happy ending" that is given to comedies and there are definitely a number of conflicts left unresolved.