Return flight out of Logan, need to return car 5am to catch early flight. Is there a shuttle that runs that hour? Will Enterprise be open at that hour?

Yes

Yes they are open...even if the customer support team is not available, you can return and drop the key in the counter

Yes, they are open 24/7 and have shuttle service to the terminals. It is an easy process..Enterprise is at the top floor..then take the escalator to the ground floor for the buses

Yes to both