Can you bring your own cooler in the park

Yes, but no glass.

Must be one of those small soft ones ... You can bring in water and Gatorade bottles ... All vendors will give you a cup of ice water if you ask for free

Yes small one that you can carry or put on your back is fine

I believe so. I think it depends on the size, but I think so. If you need water they have ice water cups at refreshment ports for free 👍