Deciding where to stay in Cozumel involves some trade-offs. Almost all of the really good restaurants are in the Downtown San Miguel area, so if you want to be able to walk to meals and shopping, you stay in or near Downtown. Probably the nicest Hotel within Downtown is Casa Mexicana - very nice hotel, but is has no real pool (just a kiddie pool on steroids). For a downtown hotel with pool, most of the choices are a lot less luxurious, some downright spartan, like the Casona Real (about 3 blocks back in town from the North end of the main plaza). For nice hotels with resort like amenities that are close to Downtown, easy walking distance, there is the Cozumel Palace -very nice, but expensive and All-Inclusive. And no real beach - pretty much everything from a ways north of Downtown to past the International Pier has hard ironstone shore, so whatever "beach" area you get is going to be small and man made, and you'll need to navigate the rock to get into the ocean. Then, a little further south, but still in easy walking distance to town, is the Hotel Cozumel and Resort. A very nice hotel, big comfortable rooms, and a huge pool area. It has an AI option, but I would highly recommend against it, as the food (with the possible exception of the Breakfast Buffet) is TERRIBLE. Unless you are going to drink enough to make the AI supplement worthwhile without eating meals there, don't do the AI. There are some nice Resort type places in the North Hotel Zone, but I've never stayed up there as almost all of the diving sites are way south, and very few Dive Ops will pick up that far north. However, if you dive with Aldora (an excellent high-end valet Dive Operator) you might consider the Villas Aldora, where Aldora will pick you up right there, or some of the nearer North Hotel Zone resorts, like Playa Azul, where you can take an easy bike ride to the Aldora Villas for Diving. Then there are the hotels close to the International Pier - Casa Del Mar, El Cid La Cieba, and Park Royal (that is an AI, so I've never stayed there, but I hear that it's nice). All very convenient to diving as most operators can pick you up at your hotel pier. You can still walk to Downtown for meals, but it is a hike - about 2.5 miles, more or less. Taxis to Downton are 80 Pesos (about $5.00 at current exchange rates) each way for up to 4 passengers. Then you get to the Intercontinental Presidente, about 5 miles south of Downtown, and right next to the Caleta - the marina where almost all the Dive Boats are moored overnight. The Intercontinental is the ONLY 5-star resort on Cozumel, and very nice, but expensive. And if you want to try the restaurants Downtown, its about a $10 cab ride each way - you're probably better off renting a car. The big AI Resorts are a ways farther south - beautiful big natural beaches, and very convenient to Diving, as the reefs are literally right offshore, but very isolated from Downtown. But hotels are not your only option - for a stay of a week or more, you might be able to rent a Condo or even a House for not much more than a Hotel room, maybe less than what the very high-end hotels cost. El Cantil condos are almost immediately adjacent to the Cozumel Palace, a great location very near Downtown, and very nice - check rental rates and availability on those.
As to Dive Ops, I suggest that you research the high-end valet shops that are highly recommended on this Forum - Aldora (a big operator, 6 boats, I think, so divers are grouped together based on ability and experience; HP Steel Tanks, Steel 120s and Steel 100s are standard, with Nitrox available in any tank size, so very long dives, and a long SI at a beach club; Aldora takes care of all of your gear except wetsuits for the whole trip), Blue XTSea, Tres Pelicanos, Living Underwater, etc. Just scan these boards to see who likes which Dive Op and why to figure out which Op most closely matches what you want. And check to see where they'll pick up - some will pick up at a pier Downtown, a very few may pick up North of Downtown, almost ALL will pick up anywhere near the Caleta.