This seems like a silly question, but I don't want to assume a player can't do something.
Since it requires a "hit roll" it appears to be eligible for crits.
Would this mean if a Fey Sorceress scores a critical hit with her laughing touch attack, the opponent is stuck laughing for 2 rounds instead of 1?
No, because "A critical hit means that you roll your damage more than once." Laughing Touch does no damage and there is nothing in there about extending the duration. A natural 20 just means that you hit in this case.
I would say so, but I'm sure others would not. If this turns out to be true, could you take improved critical laughing touch?
No, but you certainly can take Improved Critical Touch Attack.