Where is there a parking where I could leave my car to be able to walk across the bridge?

Couple of garages nearby as well. You can book via app to find cheaper rates.

There is but you must get lucky to find a spot. You have better chance of doing it from the Brooklyn side than Manhattan side.

Yes, but it's extremely hard to find free spot. Usually if I take my guests there I sit in the car and wait for them. You can of course park your car in the nearby garages $$$

Parking

Yes