I assume you're staying in Manhattan? If you stay in central Manhattan and the Village you should be safe wearing headphones, but I wouldn't recommend it.
A can of soda cost $1 or $1.25. Some with a bag of regular chips.
There are a ton of museums, and Broadway is pretty nice.
There are a ton of bodegas around in Brooklyn and you can usually find a deli or small store like on the OP on almost any block.
No drinking wine under 21 is not allowed.
J&R used to have a good computer section but they're stuff now.
