NYC Tourism: on carriers, attractions and food

Dear Vergers (?),

I'm going to travel to NYC in July and I'm planning currently my trip. I think that there are a lot of New Yorkers here on The Verge and I would be glad if someone could answer those questions:

1. I'm going to stay 3 days in Washington, DC and 8 more in NYC. Which prepaid plan/carrier can I get? I'm going to travel with an unlocked iPhone 5, and I'd love to get LTE.

2. Where can I eat great inexpensive food in New York (preferably Manhattan)?

3. Can anyone tell some off the beaten track attractions in NYC?

4. What's the best way to get from DC to NY? Train or plane?

5. I know that B&H is the single greatest store for cameras, lenses and stuff like that. But I've heard that they sell other sorts of gadgets. On that matter, how they compare to Best Buy and/or RadioShack?

I've never been in the US before, so some "n00b" info would be also useful.

Thanks for your help! If you've tips that aren't related to my questions please feel free to comment them! Sorry for the rather poor English and thanks!