Can you exit and re-enter the same day without paying the $5 fee twice? We are staying near by and have small children that need naps.

No doubt

Yes, they give you a receipt to put on your dash. Just show that as you re-enter.

Yes you can enter and exit park as you wish without paying $5 fee more than once. Please keep in mind there are two tolls to get to the park. One for a dollar and one for 75 cents. Park closes right after sunset.

Probably.

It's a day pass, so ideally you shouldn't pay! Not sure though.