OK. First and foremost, make sure you are prepared to do lots of walking.
As the local Queens guy I recommend you check out MoMA PS1. http://momaps1.org/ They are Closed Tuesday & Wednesday.
It’s 3 stops on the 7 train into Queens from Grand Central. If your daughter likes the exhibits they are currently running it would be worth the trip. Outdoor walking space with sculptures as well. M Wells has a little dinette restaurant, they are a hot local chef team about to open a new space in the 'hood. Take a walk back to the heart of LIC, Vernon & Jackson Aves. There are more cool restaurants, coffee shops & small shops, and a beach/park on the East River with great Manhattan views. It will undoubtedly be less crowded than midtown. If you’re interested in that I can feed you some further suggestions on the restaurants & stores.
Central Park is walking distance and you can walk up in the park to the E or W 70’s for the museums, or bike but I don’t have a solid suggestion for bike rental in the park. There is the Citibike program, you can look into that. http://www.citibikenyc.com/stations Rouge Tomate has their restaurant at the southeast corner of the park on 60th and food cart at 64th by the park entrance. Vegetarian/vegan options at both.
Here’s the search for ‘vegan midtown east’ on Yelp
Unless the place has a meat focus, most restaurants have some type of basic vegan/veg options like salad. Kajitsu is a top vegan place so if you want to do one nice dinner that would be it. Might be too much for the youngest. Health Nuts is a great place to know about for ‘emergencies.’ I haven’t been to the one on 45th & 2nd but assuming it’s like the one in the 60’s they have all organic groceries, snacks, drinks as well as a juice bar, deli, and hot foods as well as some seating. It would be a great place to grab things to keep the kids happy or for a quick and easy lunch. There are a couple other places listed I don’t know about, but they look good. There’s a Grimaldi’s on 26th St & 2nd Ave if you don’t want to trek to Brooklyn for pizza.
The High Line is a great park on the other side of town. It’s worth the trip, if you start at the top (23rd & 11th Ave) and walk south you can end up at 14th St or below and work your way back across to Union Square & the Village. For cheap eats, check out Mamoun’s Falafel. There’s one on Macdougal in the W Village and on St. Mark’s in the E Village. Also Cafe Rakka on 3rd St/Ave B is a great falafel find if you end up down there. At the falafel places you will be able to feed 4 for about $20.
For Photography, the International Center of Photography is very close to GCT. Never been but have heard good things.
Definitely hit the Met right now (art, not opera…though the opera is good too!) The Met currently has a William Eggleston photograph and a Punk clothing/fashion exhibit. The Met in general is less crowded than the MOMA since it is enormous, though the punk exhibit is probably pretty busy.
There are bike rentals at Bike n Roll at the columbus circle entrance to central park. There are various other bike shops nearby that will rent bikes, but I always see people who look unhappy waiting in line to get a bike…so stick with the Bike n Roll. There are also the illicit bike rental folk who lock their bikes up a few blocks away (ie in front of my building!) and presumably give you a better price…but let’s just say my dog loves those bikes!