All standard fishing rules apply. License (although I've never seen someone with one), no game fish (tarpon, bones, permit), no turtles, Nassau grouper has a season. No reserve areas. Not sure if it's a law but we've always stayed 500+ ft away from dive sites. No SCUBA and spearing, freediving only.
As for your gun, you can bring it. They may want you to pay duties unless you convince them you're bringing it back home with you. You will find JBL replacement parts. For other guns, you'll only find rubbers and tips.