I see most people are suggesting museums, so I want to chime in with some other things. I'm personally not a museum-person and although I have been to NY many times now, I haven't been to a single one of them.
My first time in the city was 7 days, and I did all of the touristy things minus museums. I would recommend doing the Statue of Liberty and World Trade Center on one of your days where you have an evening show. Go down there first thing in the morning so you don't have to deal with excessive lines (I believe the first ferry is at 9, so be there by 9), and do the audio guided tour of the Statue of Liberty, and stop at Ellis Island on your way back to Manhattan (the ferry stops automatically but you can choose not to get off). (Ellis Island probably qualifies as a museum actually).
Once you get back to Manhattan walk around through the financial district and by Wall Street (Hamilton's grave is in the graveyard at Trinity Church, which is right there too). And then go up the One World Trade Center. It's pretty pricey, but the whole thing with the iPad is really well done and gives you a great (over)view of the city.
I spent a lot of my other days wondering through the various neighborhood of New York City, which all lend themselves to a few hours in the morning on your 2-show days. Same goes for the High-Line that some others have mentioned. Walk from North to South and grab something to eat at the Chelsea Market at 14th St, then take the subway back up to Midtown. Then there is the Flatiron Building, NYC area and Washington Square Park, the Upper West Side ect.
You can split up Central Park on 2 days, the lower part (Mall, Fountain, Bridge ect) on the first, the upper part on another (the castle, the reservoir...). Don't miss Bryant Park, that's something you could even do between shows because it's right there.
Another really touristy thing to do is to walk across the Brooklyn Bridge. Take the subway out to Brooklyn, check out Dumbo, and then walk back to Manhattan. Definitely do it this way around, because you want the views of Manhattan as you walk across the bridge!
I second suggestion of Top Of The Rock over the Empire State, but if you want to do both, maybe do one on a future trip and also do one at night! They are open late you can easily go up after a show!
I don't know if you're into movie filming locations, but if you are that's a fun thing to check out. They have bus tours for that, but I did some research (and bought a NY movie locations travel guide) and walked by the ones I cared about when I was in the area.
Since you love theater I can highly recommend the Lincoln Center Tour. It was so interesting and you get to see all the theaters (if you're lucky including the Met) from the inside. Check their website for info on tour times ect. Radio City Music Hall also offers a tour. I didn't do it because I've been in to see shows, but it's stunning and I'm sure it's super interesting as well.
You can visit the UN, which is something I haven't gotten around to yet, but that's on my list!
I can very little advice in hotels, I have always stayed in hostels to save money. But my friend liked the Edison when she stayed there over the Paramount next door (she stayed in subsequent years). They are a similar price, but the rooms at the Edison were bigger and the wifi was included. It was not at the Paramount.
Another tip is to get a weekly subway pass. It'll be good for 7 days, and then you can use that metro card and refill it with money (instead of time) for the last 2 days, so you don't have to worry about getting a ticket every time.
