Can you bring your own table top grill or griddle that runs on propane?

Yes you can

I'd call this number - (850) 245-2157 statewide info for Florida parks. I would imagine yes, haven't heard of a place that doesn't allow tabletop ones yet. Probably wouldn't need it though as the park has plenty of charcoal stations.

We didn't grill out while we were there so tbh I'm unsure on that one

Yes

Most likely not. They do however have charcoal stations you can use.