If I have a fleet carrier w/o the shipyard module (just the "My Stuff" one that comes with it), can I deploy a pre-built ship there? Or do I have to go to a real Starport?
you as the owner of a FC, you have shipyard, outfitting and repair, rearm and refuel access even if you don't have it installed. those modules are really for OTHER people. So short answer, yes you can.
If I have a fleet carrier w/o the shipyard module (just the "My Stuff" one that comes with it), can I deploy a pre-built ship there? Or do I have to go to a real Starport?