You're going to want to talk to the NYC Health Department. I suspect it will not be possible to use a standard apartment kitchen; they're probably going to want to see some commercial equipment, but they'll know a lot better what the regulations are for NYC than I would.
Not sure if zoning will actually come into it -- but you should probably check with somebody at City Hall to make sure.
You'll want to make sure you've got some significant liability insurance if you're dealing with any sort of food-related business. All it takes is one person to get sick and blame it on your sandwich and you could find yourself staring a big ol' liability square in the face. Do NOT go into this without adequate insurance.
Your best bet might be to consult with a lawyer familiar with small business startups. For a lot of your questions, the answer will be, "it depends on your local regulations and laws." A local lawyer who knows the ropes of starting a business could be your new best friend.
--Torka