Cabs are exempt, you don't have to use a car seat in them. If you do want to, you have to check your particular car seat and figure out how to buckle it in without a base. I haven't gotten that brave yet, since DS is only 4 weeks old and not really getting out yet.
You might get more hotel info on the travel board. Honestly, I have no idea, I don't stay at the hotels here. Food & Diapers, like pp said, any drugstore, CVS, RiteAid, Walgreens, Duane Reade. Seriously, you shouldn't have to walk more than 2 blocks in any direction before you run into one. BRU has a location in Union Square, but why bother, the extra cost at a drugstore will be worth the cab fare to get there.
You could probably use a lightweight stroller, like a MacLaren. If you & baby are comfortable you could go with a carrier and that would be fine, too. Depends on how much you like to keep the baby in the carrier and how strong your back is! ;-) Don't try to carry in your arms, that won't work, use a Bjorn or something.