Yolo, I could live more cheaply in Gainesville than I could in Cozumel at least for food and rent including utilities. Miami would cost more and Key West would be off the charts. Of course Gainesville and Key West too, have decent public transportation and I have lived both places comfortably without a car. A nice one bedroom for in the $400 a month range, utilities cheaper, food except fruit and vegetables cheaper. The one winter we spent in Gainesville, we had a sweet heart deal from a friend, paid $800 a month, but that was for a place that was furnished with all gourmet cooking needs and the place was filled with art and antiques. Two blocks from the main bus central and could travel efficiently all over town. It is that car ownership, essential many (most) places which really impacts a monthly budget, the gasoline the least of your worries.
Cozumel rents will be all over price wise, it depends on what you are willing to accept and the area. Central locations have some big pluses, walking and/or easy colectivo transportation. I like being in the fringe of downtown if you can get lucky and find a place. Colonia 10 de abril, is convenient, but does have some negatives and then there is Adolfo Mateos, bordered by 30th & 65 and Juarez and calle 11. No place is perfect, but these areas would have a convenience factor.
For me personally, I have a real problem with mildew and mold, I can't take it. I have gone to check out some places that looked good online, open the door and UGH, no thank you. Some places could be aired out, cleaned, while others are beyond hope. It would be real nice to have one room (bedroom) air conditioned. A consideration in furnished, is just what is a bed like and is it a typical size, a double which I consider appropriate for two kids, not for adults. Keep in mind when places are furnished, that refers to furniture only, no linens nor kitchen utensils, no plates, pots nor pans. Do you want to have television (cable) and besides entertainment, TV can be a Spanish learning tool (I learned a lot of verbs reading Spanish subtitles). In my opinion, $8,000 pesos monthly would be a decent price of which you should expect a higher quality of furnishing and assorted comforts. I'd expect kitchen utensils with some appliances. A nicer and larger bed, I'd plan on buying new pillows. TV would be reasonable and in that price, I'd hope for a decent air conditioner. For $8,000 you might find fancy and you might find a "you have got to be kidding".
There are a few places that have intelligent landlords that make for good value, are always in demand and rarely become vacate. There's a couple of places, that might be OK for a month or maybe two weeks that you could base at while you looked for something better suited. Look where you think you want to spend your time, stay close to busier colectivo routes and/or being able to walk. I lived for a while on 15th, close to the airport road, I could walk out my door, usually get picked up on a collective and be inside Chedraui in seven minutes. If you are in the boonies, public transport will be infrequent and you'd need to pack a lunch if going by foot. For convenience, it is well worth spending an extra $1,000, $1,500 pesos a month to have a good location. You can factor in more taxi fare with cheaper rent and a greater distance, but there are times it can be damn hard to get a taxi, even if you call for one. And when it is busy on the island, taxi base just takes the phone off the hook.
That is not a good idea "helping out" or volunteering as you could call it. That is the fastest way for you to get deported and your dive shop friends could get in major trouble. Sometimes it is just the appearance of some impropriety. Trust me, it will not go unnoticed and even if below the radar, it takes only a slip of the tongue and you'll find yourselves in a locked room at the airport waiting for the next plane. It would be close to impossible to get work papers to even be an assistant on the dock, the dive industry being the most regulated and for a long time, you needed something major, like fluency in three or more languages for even consideration. There are far too many Mexican dive professionals, not to mention former expats that have reached immigrado status that they can work and function as Mexicans. You probably can't imagine the number of people selling jewelry that would rather be working at diving, but diving just doesn't pay the bills, 6-7 months of the year. There might be 500 people qualified to work in the industry, any one of which might report you.