Sounds like a cruise ship style operator to me. Some of it is typical, you just have to know how to deal with it. Here is my advice:
1. Buy your own reg and computer. By it used if necessaryand have it serviced. It's piece of mind. There are so many computers out there, having your own and knowing how to use it is much safer. HOG has some great entry level regulator sets at Dive Right In Scuba for the price of used bigger name brand. Dive computers start at around $200 but can be found used for much less. An air integrated Oceanic pro plus starts at around 400. Again used is less. If you buy used and still don't like diving, you can still get most, if not all, of your money back out of it.
2. I love having my gear set up for me and my wife.....but we still check everything.
3. We research before we leave and only dive with "6 pack" boats that take no more than 6 divers at a time. It's common in the industry.
4. Especially on the first dive, there is the urge to stay next to the dive master. If it gets crowded, just hang back or hang 10 feet above. "Stay with the group" does not mean everyone needs to be in a 20 foot circle. As long as you can see the DM and he can see you, all is good. As far as the wildlife issues, some Dms will what you saw, some point it out and swim by, some are like a biology class stopping at every little thing.
5. Assholes, idiots, and clueless people are everywhere. You will find them even on dive boats. Just a fact of life. Not much different than who you get stuck next to on a plane.
6. It sounds like most of the issues were operator based. The location specific areas of this forum are a tremendous resource to determine who to dive with and who to avoid. Use them, we are here to help.
7. Dive locations are different in their individual personalities. Cozumel is famous for its "babysitting" style of diving. It's great for new divers and sometimes bothersome for more experienced divers. We love it...love everything about it from the diving, food, people, cost, etc. we have dove with several operators down there and never experienced any of the issues regarding the operator side of your problems. But see #5. I have been run over, kicked, etc. it happens, and I have kicked a couple of people as well on tight swim throughs. I always apologize back on board.
8. Computers can be set for so many variables, there is no telling what it was beeping for. I love my Oceanic Pro Plus and it always tells me i have come up too fast. Just grabbing it and pulling it up to see can trigger that alarm.
9. A private DM in Cozumel is $50 for a 2 tank dive. Totally not necessary, but an extra comfort margin for a new diver. On our first trip, We chartered our own boat for 4 of us and it was $15 more per head to do it than a regular dive boat. We were on a cruise and knew we did not want to dive with the ships company, I was worried about getting back on time. If your on a cruise, I would advise against booking with the cruise ships dive company. There are just too many better and cheaper options out there.
10. Your experience can be typical of larger dive operations. Smaller, service oriented, operations are easy to find in this forum and most of the problems will not be an issue....again...see #5.
Before your next trip, use these forums. I can't recommend Cozumel enough for new divers. Message me if you want my preferred operators. Good luck and safe travels.
Jay