Is it possible to take a picnic lunch and either eat it there or leave the museum and eat somewhere else then get back into the museum afterward?

You need your hand stamped that way can reenter the museum without any trouble

You can reenter the museum as long as you get a hand stamp I belive. I don't think they allow outside food and drink inside, but there are plenty of places around the museum for a picnic.

Just outside there are areas where you can. I may be a bit hot but across the street there is a shady park. It is cute.

I do not know as I was there for a special event. Sorry.