Another option for your family would be to move to Forest Hills, Queens, which is 15 minutes from midtown via the Long Island Rail Road. There are three excellent elementary schools, all within close walking distance of the rail road station: PS 101, PS 144, and PS 196. All three of these schools have extremely strong academic programs, as well as excellent arts programs, and regularly place children in Hunter, and the top selective middle schools in the City. You may find good articles about living in Forest Hills in Time Out NY City Kids and The New York Times. As for the schools, check out their listings on InsideSchools, as well as the school websites.
I live in Forest Hills (former home of the US Open, by the way), and my son attended PS 196, and went to the G&T program there (a pull out program, not self contained); however, it was our experience that the school was so strong that probably at least 20 percent of the study body would be considered G&T these days.
The Forest Hills community is extremely child friendly, with lots of parks and playgrounds, and you can walk to everything (movie theatres, restaurants, shops, etc.) There are also great afterschool programs at a variety of local Ys, community centers, and also at 101 and 196 (not sure about 144).
Good luck.