Most art galleries carry insurance on the pieces they hang or have in inventory, usually against theft or fire. However, that doesn't mean that ALL art galleries carry such insurance.
It depends on the gallery... Some galleries have studio space attached and, given the artists permission, may be used as gallery space for that artist as well. People like seeing into the "behind the...
All art galleries store art work and are always circulating pieces. The usual storage space is a room in the basement. At the Louvre you can see some of the storage as you ascend the first flight of...