Is there a public restroom? If so, where?

There are a few, one set of bathrooms is off the main parking lot back by the manor house, just past the playground. Just walk past the fountain shaped like a tree and theyre on your left

Yes, one by the playground and one by the white building called the stables right by thr parking lot. Just follow the curve around through the park entrance

In addition to the locations already shared, there is a restroom in the white building across from the old schoolhouse.

In the main parking lot at the stable area. You cant miss it

Yes