Is it possible to customize NetSuite so that when billing from a PO, lines can be checked to bill vs. manually removing the lines?

If you turn on advance billing and receiving then PO lines will reflect billed and received qty automatically. No need for manual line close

Is this part of SuiteBilling or does that replace it? I don't know a lot about that module, but I do know we have it and aren't really using it.

That's standard one in enable features