Once we enter the museum can we leave, go to lunch and them come back into the museum?

Yes, this museum is a smithsonian so admission is free so enter and leave at your leisure. You may though have to wait in line for entrance depending on when you visit.

Yes, you can. Even admission is free.

Yes you could. Considering there is no museum entry ticket/ fee as such, entering multiple times in a day shouldn't be a problem :)

Yes. Its free. You might have to do the line again but you can go in as many times as you want