Yes, all of SJU is connected airside. It is a long walk from AA world to the other carriers' gates, or vice versa, but you don't have to reclear security. For refreshments, the non-AA gates have the better value concessions.

The good news about the luggage is that other passengers will not be able to leave with it, since luggage is checked against claim stickers as passengers leave the terminal. At AA, it would be probably be pulled off the carrousel and left at their Baggage Services station. Can't say what other carriers would do, but presumably something similar. There are many reports of theft from luggage in SJU. Transiting there 4-6 times each year I have only ever lost luggage tags, which have inexplicably been removed on two occasions.