Definitely do all wiring before you close the walls and always have extra/future circuits pulled to a location that is easy access for future purposes. It’s also good practice to test all power before closing walls. Like mentioned above 20amp breakers require number 12 wire...15amp breakers require number 14 wire. Try to think of all your lighting ideas and how you want to set them up before closing walls(switch placement, light fixtures,fans). You might also want your plugs to operate on a switch or multiple switches. Good luck and if you get overwhelmed hire a electrician.

If you plan on a sub-panel for your game room you definitely want to run a 220 4 wire system to it.

I would do 2/20amp circuits per side...5pins per 20 amp so a total of 4 breakers(20amp) from your box.

All your lighting circuits should be separate from plug circuits and do not share the neutral(white) with your plugs...always keep your plug and lighting neutral wire separate.

Yes you can probably load that with a 15amp breaker.