I love NYC. There are so many amazing things all around the country and world to explore, try not to get so locked into Disney-think that you don't enjoy other destinations.
What you should plan to do really depends on the interests of your group, especially kids. Is your son the only kid going, or do you have other children? What kinds of things does everyone like? Is he a sports kid, music kid, science kid, etc., or a combination of several? If you can give some more info on specific interests of your group, I'm sure we can help with more ideas for you. I tend to lean toward theater, but I'm sure others can help out with other areas.
For a show for a young teen, Beetlejuice, Harry Potter and the Cursed Child (keep in mind that there are parts 1 and 2), King Kong, Oklahoma!, and Wicked seem to be pretty good options. You can try to get tickets at the TKTS Booth in Times Square or one of their other locations, but that doesn't allow you to pre-plan, as tix are only for that day or possible the next, depending on which location you use. Personally, we usually just bite the bullet and order tickets online from the venue's site. It costs more, but we know exactly what we're seeing. I usually just Google the name of the show and 'Broadway" and look for the official site.
I haven't seen any comedy shows that I could recommend (Drunk Shakespeare was NOT for teens and was absolutely terrible anyway) and I've seen the drag show at Lips twice which was fun but again, probably not really an optimal choice for a 13 year old, lol.
Good luck, and have a great trip!
My son is 13 and really not into anything. He likes funny, and to laugh, which is why I want to find a comedy club.