I doubt it, it's not really that big, and there's a lot of roped off areas.
Nope
To the pond behind it yes plenty of room to run around
There isn't a dog park section, but you could walk one around the perimeter.
It is really not that formal. Outside the park in the picnic area is probably fine but where the digging is happening, which is a petty small area, I would say probably not. It gets pretty hot so don't leave your doggie in the car! I don't remember seeing any signs so a call might be a good idea.
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