Can you put a tent outside of a cabin you ate renting?

Yes. You can.

Yes

Yes I have seen people do it there all the time ..

It's very specific to state parks. Advisable to call them on their number on the website to confirm it.

Yep.

I stayed in the rv section and was only there 1 night. But most Texas State parks let you set up a tent next to or close to the cabin, at least the ones I've stayed in do. I would contact a particular park for specific info.

Depends on the location but am sure you can, for the kiddos