1) I bought him a new Squidgee ball yesterday and he loves it. He gets excited when he sees it. He plays (chews) rough with it and kind of grunts while he does this. Is this normal?
This is COMPLETELY normal
2) He likes to play rough and nip at my fingers. He nips at them almost all the time. Today he nipped really hard and left a little bruise. Is this normal? How do I stop him from trying to nip at my hands? He has a few toys that he likes to chew, but he seems to especially like my fingers.
Don't let him bite or nip at your fingers. You need to discourage this behavior. Whenever he tries to nip at your fingers, tell him NO and give him something else to chew on.
3) How do I know if he likes me and is happy? He does lick my nose and my fingers, but I don't know.
If he is licking you, that means he likes you! 
GOOD LUCK with your new baby! And WELCOME to YorkieTalk!