When I was in Florida, I rented an electric scooter because I need to baby my knee. It was definitely less expensive to rent outside of the theme parks vs within each them park. They have some models that can be broken down easily enough to put into the trunk of a car, as long as you are under 200 lbs. However, The problem with motorized wheelchairs in NYC is that most curbs are not wheel chair accessible and it's definitely hard to manipulate in crowds. Forget about using the chairs downtown. since some curbs are higher than others. Also, hotel rooms in NYC are small ( at the 3 star level in tourist area) and it might be difficult to keep it inside the room and move around it.
Book of Mormon is the hot show, but it's a little risque for some people. Avenue Q is an older show that has ticket available during most week days, it's a risque puppet show.
If you don't eat much, I personally like street food. Stuff like Gyros and falafels are inexpensive. The gourmet food trucks are pricier. There's an gourmet empanada truck in Times Square in front of Toys'r'Us - $10 gets you 3 small empanadas and a bottle of water. There's lots of green grocers where you can buy food by the pound. While a sandwich can be expensive, some of them are large enough for 2 meals. You'd probably save money by not eating at the hotel. You can also pack some bottled water and just bring one with you during the day.
As far as lotteries, each show is different. A group of us got lucky and 2 of us got our names pulled the 2nd time we tried for tickets. The problem with the lottery tickets is that they are usually for the 1st row and in some theaters that = no leg room and having to tilt your head back during the entire production. It is thrilling to win them, but tall people would have a really uncomfortable time sitting in the seat for the length of the production.
I haven't looked at shows in a while; so, I'm not sure what's playing right now.
Personally, I wouldn't put Lion King at the top of my list. If you've seen the movie, all of the songs are too familiar. The puppets are the most interesting part of the show, but if you've seen them before, then it's not as spectacular. Yes, I did see it, but after seeing the movie and after seeing similar puppets at a different production. Everyone I went with felt a little let down, but it's based upon our prior experiences.
It was Kristen Chenoworth and Idina Menzel who starred in Wicked. However, they have long since departed the show and gone onto other things. Right now, I don't think there are any big names in the roles, but, as others have said, the show should be more about the show vs who is in it.