Yes, they even have storage. Old Mother Hubbard Cupboards near the front has shelves for coolers/other stuff with picnic tables. Not locked but we've never had a problem.
I saw many families who had brought their own food ans drink in coolers when we were there yesterday.
Yes. We left ours in the car and when we were ready they stamped our hand so we could reenter
Unless people had strollers I didn't see coolers in the park
Yes, you can bring a cooler. The type you mentioned is fine.
Yes outside food is ok. I brought a cooler full of drinks with me
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