I was there and if you like fine dining in a nice restaurant, try the Russian Tea Room.
http://web.russiantearoom.com/
It's definitely quite a spot if you have the extra cash. Their wine list is like a book pretty much. Food is awesome and it has very nice ambience (IMO)
There's all sorts of restaurants down south (near where the WTC used to be). They were mostly Italian and I wish I had more cash and time when I was in NYC... I don't remember which Avenue they were on...
Shopping wise, I'm not a big fan of shopping so I can't really say.
As far as underage drinking goes... I guess it does depend from one bouncer to the other and also on the bartender. Why don't you buy some Duty Free in the London airport before you come?
It ain't gonna be as cool, but it should give you enough a kick
I do agree that nightlife kinda sucks before you're 21 (I'm not 21 either). There are places that you can drink that's for sure.