While playing the game I noticed that if you beat Fun with plushtrap on the menu and then start a new game you get a 2 hour head start on night 1. Is that allowed or not?

I would say yes, and would start the timer on starting the Fun With Plushtrap game. The thing is, you need to be fairly good at the game for it to save time with fast nights on. If it takes you more than 30 seconds, I'd say it isn't worth it. But I know you are good at these games, and I bet you can do it well.