Disclaimer: The below is based on my sole experience at Phoenix Comicon. It may or may not apply to NYCC.
You can bring a backpack. The wheeled restriction is because space will be at a severe premium.
Seriously. It’s going to be packed. Jammed. Crowded as hell. Phoenix Comicon was bad enough, and it’s relatively small compared to the San Diego and New York cons.
You’re welcome to bring in anything you like that’s not on their restricted item list (which consists of wheeled bags and weapons, mainly). Security isn’t necessarily going to be watching for theft, that’ll be up to the individual vendors. Your own food and drink shouldn’t be a problem, and I expect that the food court employees and their customers will be grateful for one less person in line.
You should be able to leave and re-enter, though it’s possible this specific location will have different rules. At Phoenix Comicon, the pass was really only checked when I entered the exhibition hall where the vendors and guests were; the conference rooms were pretty much open to all. Either way, a day pass should be good for the day, not just one entry.
Recommend that you do as much as you can in one trip, though; if you leave, you’ll have to requeue at the entrance, and I very much doubt the line will ever become short. It’s very likely that after you go through it once, then stand in various lines inside, you just won’t want to return if you leave.
General tips: Pack water, a camera, anything you want signed, but try to leave space if you plan on buying stuff. Swag takes up an awful lot of space real fast. You’ll probably want to eat before you go or else bring snacks because it will take an eternity to get food from the food court.
If you want to see panels, plan out which ones you’d like to see and get an understanding of the schedule and where they’ll be. It doesn’t look like you’ll have to sign up ahead of time, but you will want to be sure you have time for the stuff you want to see.
Celebrities will probably have schedules available telling you when they’ll be at their booths for signing, photo ops, and the like. Find out ASAP and make a note to be in line for that. The popular guests (and Miss Day is pretty popular) will have a lot of people waiting for them, and if you get in line too late you may find that she’ll disappear for a panel or something just before you can get to the booth.
Basically, the biggest thing to expect is that there will be people. A lot of people. A whole friggin’ ton of people. So many people. Your activities are going to be primarily based around navigating the sea of people to get to where you want to go.
If you have the patience for that, of course, it can definitely be a lot of fun. The cosplayers alone are worth going to see (they’re why you want your camera, not the professional guests).