The Game has to start right away or I can use the button has a "PlayButton" in the Menu for the game to actually begin.

I ask this because the rules said that the button can only do one thing, but I would like to know if that counts for the game Menu as well.

you can have main menu's you just need to have the fun button be in the core gameplay

yes, but you don't have to use the FUN button to start the game, you can also have a normal main menu (like Malice said, the limitations are restricted to the gameplay)

Thank you both for the answer, really appreciate it.