Apple Care doesn’t cover liquid damage. Apple Care+ does. If you do consider it go for Apple Care+. I am glad I got it when I got my Macbook Pro. 8 months later accidental liquid damage at work. With Apple Care+ cost me 299$ without it 1700$. Everything got replaced and from start to finish the entire process took 48 hours. You never know what can happen no matter how careful you are. Is not always your fault. Accidents happen.

I doubt that you can use it with Apple TV. Only iPhone has features that works with Apple TV

I don’t turn off my Mac but expect when I update it I don’t remember seeing the loading bar. As far as your warranty you are covered based on the date of purchase listed on your invoice no matter what the system says. Customer service might be able to adjust that on the system based on your invoice but I don’t know for sure, you can ask them. Either way you are covered for 12 months. Don’t lose your invoice/receipt.

If you want you can purchase Apple Care in the first 60 days without any inspection or visit to the Apple Store. You can do it online or from your settings menu from your Mac.