Can you leave and re-enter a few hours letter after paying admission price?

Yes you leave the Library and return once you have your ticket stub.

When you get your ticket you get a clip that you can attach to you clothing. I'm pretty sure its for that reason so you can get back in, but I'm not entirely sure and I have never done that before.

Make sure you save your receipt, but this is something you can do.

Yup. Re-entrance isn't a problem.

Yes.

Yes, you can.