Once in the park, can you exit out to make a quick run to a store and return without being charge the entrance fee again?

Certainly, you either have a pass from the entrance self service area, or you get one at the rangers station.

YES, as many times as u need.

Yeah u get a sticker for the day or nights u are there for

Yes you fill out a form at the box, keep your copy and drop payment and registration into the little box

Yes, but keep in mind there isn't much near by. The Bend General store you pass on the way in only has the most basic supplies, anything more and you have a 30 to 45 minute drive, one way, ahead of you. My advice is to plan ahead and take what you need, the drive is too long.

Yes, you get a day pass for however many days you're staying.

Yes you can

Yes