What happened with Stu & Jade starting a relationship at the end of the first film? Simple, he mentions at the start of the sequel that he's still seeing a therapist to cope with the events of the first film. Stu simply couldn't date Jade because she & Tyler were reminders of what happened in Vegas.
Phil initially comes off an obnoxious frat boy, in contrast to Stu as the responsible one. However, once the reality of losing Doug sets in, Stu basically has a breakdown & Phil takes control of the situation whilst calming down a distraught Alan. Phil's a married teacher with a son of his own, so it makes sense that he's the one who becomes the voice of reason rather than Stu.
Doug was severely burned after the guys found him, to the point of his chest hurting like crazy when somebody touches it. Then in the second movie, from the prostitute's story, we can tell that Stu, as Alan puts it, "has semen in him". Additionally, he's been treated with a needle from a not-so-reliable tattoo artist. Now, Stu is a responsible and caring guy, when he's not stoned. It's safe to assume that he'd take all kinds of tests before endangering his new wife. This means that neither of the grooms had sex on their wedding night. And you thought that (missing fingers aside) there had been no consequences!
What if they'd found Doug straight away & left without resolving the issue of the tiger in the bathroom? What if they never found the baby either?
Jade does have a key, and was only out of the room a few minutes to get coffee. She would have taken her baby if they had left him there. The tiger was locked in the bathroom and so wouldn't have been able to eat the child. If they had left the tiger there, one can assume the hotel would have called animal control and it would eventually be returned to Mike Tyson (assuming it was legal for him to own it in the first place).
The stinger at the end of part three...just stop for a moment and realize the unavoidable truth. They were not the only ones to eat the cake.