Unless you are really attached to using your own equipment you really need not bring it along. Save lugging it around everywhere on your trip, paying extra for the insurance on the plane, and tending to it by using the very good rental equipment at the diveshop. It really does not increase the cost much at all but the freedom is wonderful.
I agree with the above poster, Paradise Down is a top notch shop. They are PADI and have a very safe operation. Their new boat, a 45 foot divers dream, gets you to the outer atoll dive spots quickly and comfortably. It even has a head. Their staff are wonderful, quite accomodating and just plain good people.
As for nightlife, try the Oceanside bar and perhaps the Disco. Not alot of nighlife but you will probably be tired from all the diving and want to call it an early night anyway.
Caye Caulker is a great spot, hope you have a terrific vacation.