Can I bring a tent (like a mini pavilion tent) and set up in an open area on the grounds (where the grass is)?

yes but no grilling yur welcome ms furrow anytime

Yes, that is allowed.

Thanks so much! I actually just got off the phone with the representative for Burke lake Park, and she said as long as it's not a ridiculously huge tent not much more than 10 x 10 then it is acceptable.

I have seen people do it before. I am not sure if there is a rule against it.