Does the place have a parkinglot for customers?

Well, stret only

No

I did not see any parking. It would be best to take public transportation, and leave the car at home

Not that i know of

No you can park in a garage near the museum but it can be pricey. We use the spothero app to find cheap parking and then take the subway in

Yes, the following is from the website: 81st Street between Central Park West and Columbus Avenue. The hours of operation are 8 am–11 pm. Rates are as follows: Up to 1 hr: $24 Up to 2 hrs: $27 2 to 5 hrs: $34 5 to 10 hrs: $44 Max to close: $49

I am not sure about that