Typically build threads go under their respective consoles, and Modding 101 is for tutorials or beginner questions.
A mod can move it for you, but if they haven't yet I guess it's fine. Good luck with your project!
I've got a Wavebird case I was only slightly into modding for a previous project (not submitted for a build contest). Can I work from there and submit it this year?
Please read the guides regarding video wiring.
The gamepad screen itself will not work, however there are screens you can buy that are the same dimensions if you're looking to use the case.
If that board is expecting 1.5v AAs, you wont be able to use it with your 3.7v 18650s. The regulator stage later won't make a difference if this board isn't suited for protection or regulation of your higher-voltage cells.
That's gonna be a no. It'll chew through AAs like nobody's business and that build quality looks shoddy at best. Invest in an appropriate Li-Ion solution, like a red board or one of the BB store BMS modules.