Can you leave the zoo and come back in on the same day?

Yes make sure you get stamped on the way in.

Yesyou can take food/drinks coolers in so you really don't have to leave if you don't want

Yup..

Yes, keep your ticket stub

Yes

As long as you get a hand stamp. You may have to pay for parking again, but not sure, id get a receipt if possible for that one.

Yes get your hand stamped

To my knowledge, no.