I just got back from Bermuda. It is wonderful!
Yes, the ship, unless you are told differently, will be open 24 hours a day for on and off.
There are lots of cars (it's not crazy busy). Remember, they drive on the left there. I did not see anyone driving a golf cart, but I wasn't at a golf course. It is pedestrian friendly. There are sidewalks. I did not venture to St. Georges or Hamilton this time. There is limited shopping at the Royal Navy Dockyard. There are a few shops. You can buy some souvenirs. I'm not sure what you are looking for. There is the Clocktower Mall and some other shops, Dockyard Pharmacy. I didn't spend that much. I shopped at two stores.
If you want to go to Horseshoe Bay Beach (which I highly recommend), you can get a shuttle to and from the beach for $7 pp each way.
Take a mask. We had to wear when we walked through the small terminal at the dock. RC will provide a mask in the stateroom if you forget one, but make sure you take it with you when you leave the ship.