@Bopsmom, sounds like you're set on seeing multiple shows so of course the Theater District is the best place to stay for convenience. I'm in the city so often that I've never done one of the Double Decker tours, so I can't comment on those, but I've heard mixed things. The Statue of Libery and Ellis Island are nice, though I wouldn't recommend climbing the statue if you are worried about overexertion for your mother. I really liked the Little Mermaid, I think you'll enjoy it. As far as other family-friendly musicals go, I'd recommend Mamma Mia!, Hairspray, and Mary Poppins. Mamma Mia! and Mary Poppins definitely have been on TKTS in the past, Hairspray may or may not be. I don't know that I'd want to see a show three days in a row, but that's just me, and granted I can go there pretty much whenever I feel like it.
One restaurant that sticks out in my head is Mars 2112. I've been there twice, the first time was an awesome experience and the second time I thought it was just okay, but it is a pretty unique place. It's a themed restaurant, and the theme is... Mars. They "transport" you to Mars on a little motion simulator ride, and then the whole seating area is a pretty nicely themed Martian landscape. Though, being Disney fans, our opinion of well-themed is probably a bit higher than normal standard, lol
. It's on Broadway and 51st, good place for kids. Not cheap, but nowhere in NY is cheap. Think Applebee's food with a 50% surcharge. And, to be fair, even if you go to Applebee's in NY the food is significantly more expensive than ones out of the city.
It sounds like you've done a few of the major tourist things in NYC already (Rockefeller Center, Radio City, Macy's, Empire State, etc.), so maybe take it a little easier this time. As above, I'll recommend the Museum of Natural History and the Metropolitan Museum of Art again. By the way, there's a lesser known part of the Met called The Cloisters, kind of a medieval architecture recreation with some great artwork on display. It's way uptown (there's a subway/bus that runs to it, but it's a semi-long ride, about an hour from the Met), but the admission to both is included in the same price, so you can do both in one day. The people at the Met can give you directions from there to the Cloisters if you wind up going there and are interested. The Cloisters grounds are gorgeous and it's really in a beautiful area of the city (Fort Tryon Park).
If you don't feel like going quite that far, there's also the Guggenheim and Museum of Modern Art, but those don't tend to be very kid-friendly. Head down to the South Street Seaport as well, that's a cool place to hang around for a few hours. Since you're in the Theater District, you can spend also spend plenty of time in the various stores in the Times Square area and also on 5th Avenue near Central Park.
As far as your flight, whether that's a good time for a flight out is really unpredictable. It depends on how delayed morning flights get- if they get really delayed it could affect flights as late as 11, but I'd say 11am is a pretty safe time for minimal (not zero) delays.