This is probably something to call your insurance and ask about. Any billing from the hospital is going to go to them... I can tell you I had one pre-delivery trip to L&D for a false alarm and never saw a bill, but we have good insurance and they pretty much covered everything.
It depends on your insurance. We have Tricare Prime (military insurance) and pay for nothing.It was all covered by our insurance, but we still got a statement. If we didn't have insurance, the visits would have been close to $600 each visit. I assume your hospital will charge you for each visit, but I'm not sure. My visits were to stop labor though, so that may be a different story.
Re: L&D ?