Yes, everything is looked at, as far as gear, a little more closely, but you can indeed get through the airports with your console and regs; just be sure that if you have a Remora type dive knife on your LP hose, that you take it off prior to walking through the security checkpoints. Someone might mistake it for a nail clipper and have you arrested, since it's easier to get through with a bazooka and hand grenades than it is to get through with a toothpick or nail file.
As to destinations in Cozumel, you simply cannot do better than Del Mar Aquatics, located to the left of the cruise ship docks when you get off the boat in Cozumel. You can ask anyone taxi there to take you to Del Mar, and you can either shore dive, or catch the dive boat (Splash, Splash2, et al) right behind the first hotel to the left of the docks. It's called La Ceiba/ El Cid, and if you ask Valerie at the desk (she's the concierge), she will hook up some diving for you.
Out the back door of the hotel, it is only a hundred foot walk to the dive shop, and you can dive immediately behind the hotel and see the old wreckage of a DC-3 airplane in 35 feet of water. If you're into fish, I did a REEF Environmental Education Foundation Fish Count down there in February and counted 146 species on one dive with about ten other dives, right behind the hotel. I do not know anything about diving in Playa, although it's a great night spot, and can be reached by ferry from the mainland, if you have that much time to waste.
Jacques Cousteau wrote that Palancar Reef in Cozumel is the "most beautiful place on planet Earth," and for a good reason, as you'll know, once you experience the specatacular drift dives off Cozumel. Del Mar has one particular divemaster, Pedro, and if you get on his boat, you'll see more life than you can imagine, as he takes special pride in showing his divers a good time.
Wishing you the best and safest travel!