For hotels, I would stay away from the New York Inn (rough) and The Row. The Row's location is perfect but getting in and out of the lobby and getting in and out of the elevators can take upwards of 20 minutes from your hotel room door to the street. That's some serious time when it's 7:30 and you have an 8pm curtain to get to. The front desk staff is shockingly rude and seemingly happy about being shockingly rude and they charge an exorbitant "resort fee".
I would suggest the Paramount Hotel, the Crowne Plaza, Millennium Broadway or, if you really want to save but still be clean and comfortable, the Mayfair hotel. I can usually book the Mayfair for close to 125-150 a night. The decor is hideous and makes the room feel claustrophobic, but it is super cheap, clean, and convenient. The other ones run closer to 200-250 a night. If you want to indulge yourself but still be reasonable, try Hilton Garden Inn on 8th and 48, the Westin on 43rd. Note: no real theater lover would ever stay at the Marriott Marquis since we had to lose five beautiful historic theaters and gain one ugly one for this hotel.
I use hotels.com because every 10th room is free. You need to keep checking back for prices, and do your search in the middle of the week; the prices are always lower when you search midweek.
By the way, your show list is AMAZING I AM SUPER JEALOUS.
As for non Broadway stuff to do: when I show people around New York, I like to do downtown days and midtown days.
Downtown:
Statue of Liberty, Ellis Island, Staten Island Ferry, South Street Seaport, Wall Street Bull, Federal Hall, Museum of Finance, World Trade Center Memorial, WTC Museum, Battery Park, Canal Street, Little Italy, Soho shopping, Union Square, Brooklyn, Williamsburg Bridge. No, this stuff cannot be done in one day, but mix and match them. And if you're doing the Statue, buy tix ahead of time, ignore anyone who approaches you in Battery Park, and plan to spend at least half the day doing that.
Midtown: Grand Central, NY Public Library, Fifth Avenue shopping, Rockefeller Center, Rock Center/Radio City tour, Central Park South stroll, Central Park, Times Square, Intrepid, MoMa the Met and Museum Mile (kinda uptown but still), Circle Line Cruises, Empire State Building, Top of the Rock United Nations. Again this stuff cannot be done in one day but that's the fun of mixing and matching.
Big Onion Walking tour: go to their site and see the schedule of walking tours they have on your dates.
I suggest you visit the top of Empire State, World Trade and Top of the Rock. They all have their unique attributes. Empire State is open til 2am, and if you go after a show, or better yet, after a show and then after a bar, it's nice and quiet and you get to walk past these enormous spaces that would be full of people if you came earlier in the day. I prefer World Trade observation deck in the daytime, and Top of the Rock is perfect right before a show.
Don't be afraid of the subway and remember what you really need to know is are you going downtown or uptown. The colors all have several different trains running on them, some local some express, so say you just saw a show, and you want to head down to the Village for some booze and good times. At Times Sq, you could hop on the "red line" (don't ever call it that) and since you're going from Times Sq/42 to 14th, you would need a DOWNtown train. This is where you will need to look at the map til you get the hang of which stations are express and which are local. Usually, all trains stop at 14, so you could take the 1, 2 or 3 (red) trains downtown. But if you want to go another stop, to Christopher, then you could only take the 1 train.
One special place in Manhattan I love is the former Ladies Mile along 6th Avenue from 14th to 23rd. It is full of beautiful, gorgeous old 19th century department stores that have been lovingly kept up over the years. And one of my favorite pizza places in the city is right at 14th and 6th, Slice & Co.
Don't feel bad if you spend one or two (or three) mornings lounging in bed instead of lining up early for more tourist stuff--you are on vacation after all.
I know some people are saying that you should save money by staying away from Times Square but I caution against this. There is nothing as heavenly as having a long day on your feet seeing tourist site after tourist site, grabbing some dinner, then collapsing on your bed, setting your alarm for 7pm, hop in the shower, be ready to go and 7:30 and casually stroll to your theater with time to spare. If you have to add walking to the subway, waiting for the subway, riding the subway, walking out of the subway doing all that hotel stuff, then repeating that subway stuff.......no the cheaper cost is not worth the aggravation.
And finally, the number one, super important, more important that all other suggestions combined for visiting New York
BRING
COMFORTABLE
WALKING
SHOES.
You will be walking more than you ever thought possible. I don't care if you take a taxi or uber everywhere. You will be walking walking walking walking walking walking. It is crazy. And great. and insane. And terrible on your feet. So bring comfortable walking shoes.