NYC is huge - I think it could help to go in with a plan. What are you interested in: food, music, art, neighborhoods?
To answer some of your specific questions:
1. Download Uber. It'll be the simplest way to get around. I still recommend the subway as a unique NYC experience (and the fastest way to get around - depending on where / when you're traveling).
2. Even within food, there's a lot to choose from. There are old NYC landmarks (Katz's deli, Barney Greengrass, Serendipity, Nathan's, Lombardi's), some "hidden" gems (the Burger Joint at the Parker Merdien), amazing high end places (Eleven Madison, Brushstroke, etc.), new hotspots (Le Coucou, Babuji, )
3. There are a lot more gluten free places in NYC these days - or places that have GF options. I'd make a list of places you're interested in and then check if they have GF options. Most places have menus online.
4. Drinks - not sure if you're a cocktail, beer, or other, but there are a bunch of beer gardens (the Standard, Astoria, etc.), rooftop bars, and high end cocktail places (the Dead Rabbit, PDT, etc.)
5. Others have suggested broadway, jazz - also great. Lincoln Center is putting on West Side Story the week you're in town - that could be fun. To me, Lincoln Center is iconic, so seeing something there makes the evening more fun.
All the suggestions so far are great. As an Upper West Sider, I could go on an on, but really, with only a few days, I think it's really about figuring out what you want to do most and then organizing around that.