We have had good experiences with Cayman University Divers, beautiful, small boat and expert personalized services.
Cayman University Divers
Boat diving on Grand Cayman is beautiful especially if you visit the North Wall and the East End, but it can be very expensive and there are some very good shore dives available - with dive ops onsite - that are much more economical so you might also consider them.
Sundivers is the op at Turtle Reef at Macabuca which is north of were you are staying. For about $15 you can get a tank and weights and have a great shore dive.
There are also shore dives with onsite ops at Lighthouse Point, Cobalt Coast, and Sunset House. Eden Rock and Devil's Grotto are popular snorkeling and shore diving sites but they get a lot of business from the cruise ships, and the reefs have gotten beaten up over the years - but they are fishy!
There is also some great snorkeling from shore, see Tsetudo's Guide below:
http://snorkelblog.tumblr.com/
Regarding driving on the left, I am not that fond of driving of it either, but I can do it and my husband actually enjoys it! But you get used to it, it's the roundabouts that get me frazzled sometimes. We have found it useful to bring a Garmin GPS and car charger with us. Garmin has good maps of the island and it is helpful to be guided through the roundabouts.
There's a lot to see and do on GC so I recommend renting a car and exploring!