Medic Diver,
I'm guessing with your creds you may be at the hospital on a traveling rotation? Good luck

A few of the hospital staff dive with Chris Sawyer, especially as their housing is in the East end and the dive shop offers a great deal for local divers. When I was there I tought a couple of traveling nurses and a few doctor's.
I'd also recommend them for instructors. They have a Course Director and two IDC Staff Instuctors on staff in addition to a number of other instructors. All have many years of experience from a variety of islands and operations. But you'll find that they've settled in at Chris Sawyer, it isn't the place that get's the seasonal help in and out, which is good. They can also teach just about any specialty you desire, although they will recommend some.
You may also want to check out Blue Island divers for some south side (of the island) wreck dives. Harley, Ian and Aitch are great guys at that operation.
As far as lobstering or spearing, can't give away my lobster honey holes as I'm back again in December but there are some good places. Admiratly dive center is the place to go for spearfishing equipment, they are in the Holiday Inn Windward passage downtown.
You'll have to make friends or call around to go out on a boat spearfishing or lobstering. The operations don't mix sport divers and hunters. It's hard to talk about all the great things to see and then have a hunter throw his line with a snapper on to the boat. Admiralty does blue water charters, free diving only as far as I know. And there are a few really great freediver hunters on the island.
Otherwise you can rent a boat, make sure to leave someone on the boat and look around. Make sure to know where the national park is as these areas are definate no take.
Enjoy,
Chris Miller
PS Deep side of Hans Lollik has some good ledges for lobsters!