Are dogs permitted in the park?

I have seen many a dog there with their owners

Yes they are under the restrictions and suggestions mentioned above.

Yes

Yes, it appears so, but please make sure you curb your dog and bring water for them as the area is large and lots of walking.

Dogs are permitted on a leash on the walking trails and picnic areas. Generally dogs are not permitted in the visitor center. Service dogs are permitted in the visitor center when wearing identification as a service dog.

I believe they are