Can we leave to grab food and come back the same day without having to pay?

Yes , but be sure they stamp your hand . Have a good time

Just ask for a stamp at the turnstiles and you should be good. So yes

Yes, you just need to ask for a stamp at the turnstiles

We brought a cooler and went to the parking lot cuz there food is spendy

Yes, just ask for a stamp when you leave. In fact, they have a covered table area right outside or you can sit in the shade under the trees in the paking lot.

Yes. When you leave they stamp your hand for readmission.

Yes, they give you a hand stamp anytime you leave the park, so you can come back into the park that day whenever you want.