If you allow your child, yes, he/she can check themselves out. I think you still have to check them in, but I could be wrong about that.
Nope. Just let them know it's less people when you show up.
I love Giovanni's Table, but you're paying for the privilege of eating at that restaurant on a holiday. So unless you really want that specialty restaurant experience on NYE, I'd skip and pay for it another night when it's cheaper.
I'm certain there will be lots of parties going on. Cruise Planner will have specifics.
I'd recommend arriving 20-30 minutes early for the show.
You can see examples of schedules in our Cruise Compass section: https://www.royalcaribbeanblog.com/cruise-compass
Cozumel: Paradise Beach is a great alternative to Nachi, if you don't want all-inclusive. I also like El Cozumeleno for an alternative all inclusive.