Yes to everything below
Yes. Just in front the museum. Very limited though. You can park elsewhere in the Fort Tryon Park as well.
There is also parking at Leaf Cafe which is a five minute walk from the Cloisters and it is a cute cafe for lunch.
Not much.
Yes, there is limited parking
Yes. There is parking.
yes
Yes there is street parking but I think it's limited, like anywhere in the city. We drove there and got lucky by getting the parking close to the main entrance.
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