I have been going to Cozumel for 26 years now, at least once a year, often as many as 2-3 times a year. Most times with a companion/significant other, often as part of a group trip, occassionally as a solo. My two favorite things about Cozumel are the diving - I LOVE drift diving, it is so relaxing and stress free once you realize that that the current is your friend, doing all of the hard work for you - and the great dining options - everything from from the higher end places where you can get a truly wonderful gourmet meal for less than $50, to the little locally owned, off-the-beaten-track gems, where a great lunch can be had for $7.00, including tip, and the Taqueria's back along 30 where you can get more than your fill of Tacos and Beer for less than $10. Because of the wealth of great restaurant options in Cozumel, I do NOT do All-Inclusive on Cozumel. I know that there are cheerleaders here and elsewhere who love the comfort of knowing that everything is paid for up front, or the convenience of never having to leave the resort for meals, and that's OK for them. It's just not my preference.
In deciding where to stay, we need to know your budget. Cozumel has a lot of options, from "stay dirt cheap and spend my money on diving and dining", which I have done several times, to "pay for a nicer room with a view and ocean access, because I can afford it", which is my norm, to "what the hell, it's only money, let's go Five Star!" , which I have also done.
If all you want is cheap clean accomodations, there are several small hotels in Downtown and further back in town that you can get for as little as $25 a night - certainly not fancy, but clean and certainly liveable. Given that you are going in August, I HIGHLY suggest that you look for one with a pool - it's going to be hot and muggy, and you want a pool to cool off. Check out Hotel Plaza Cozumel, right on the Plaza immediately accross from the Ferry Pier, very convenient to all dining locations, and with a pool on the roof, OR Hotel La Casona Real, about 3 blocks back from the Plaza, still easy walking distance from everything. You can find deals on either place for about $35 a night.
For middle-budget options, there are a lot of options, from popular Downtown locations like Suites Bahia (older place, still comfortable, no pool), Casa Mexicana (very nice rooms, only a small kiddie pool on the mezzanine above street level). I know that there is at least one Dive Op, Blue XTSea, that comes very highly recommended, that can get you dive and stay package deals at either of those places. Casa Mexicana has a very good breakfast buffet that comes with the room, and if you stay at Suites Bahia you also get the Casa Mexicana Breakfast buffet. Moving away from Downtown there are several options, but the ones I am most familiar with are Hotel Cozumel and Resort, and Casa Del Mar. Hotel Cozumel is a big hotel with a HUGE pool area, very nice comfortable rooms, and still easy walking distance to town. It has an All-Inclusive option, but, unless you plan on drinking your money's worth, I recommend against it - IMO, the food is TERRIBLE (Disclaimer - I am a foodie, and my standards are higher than most). But it's still a nice place to stay and a good location - I haven't checked prices recently, but IIRC, you should be able to get a room without the AI option for about $75 a night. Casa Del Mar is much farther south, about 2-1/2 KM from Downtown - still walkable, but it's a hike. Taxis to/from Downtown will run you 80 Pesos each way, or about $5 at current exchange rates if you pay in Pesos (IMPORTANT - ALWAYS EXCHANGE CURRENCY AND ALWAYS PAY IN PESOS. That goes for Taxis, Restaurants, pretty much everything except Dive Op and tips to DMs. You CAN pay in Dollars, but you'll pay a lot more.) I like Casa Del Mar, good breakfast, nice smaller pool area, good views and convenient to diving. One of the better budget diving options, IMO, is Dive With Martin, which is located right at the end of the International Pier, an easy walk, and DWM will pick you up right at the Casa Del Mar Pier, right across a Pedestrian Bridge from the Casa Del Mar pool.
Most of the higher end places are AI now, so I avoid them. I used to regularly stay at Plaza Las Glorias, which is no more - it is now Cozumel Palace, a very nice AI place very convenient to Downtown, but very pricy. The only non-AI high-end place that immediately jumps to mind is El Presidente, which I believe is closed for renovations, so that's not an option for you.
No, you do NOT have to dive with the House Dive Op of the Hotel you are staying at. There are all sorts of great dive ops on Cozumel. Most, if not all, will offer Nitrox as an option, with a surcharge per tank. A very few offer bigger steel tanks as standard -Aldora immediately comes to mind and is one of my favorites for many reasons. It is at the high end of the Cozumel price range for Dive Ops, but you get long, long bottom times and great service. But others here will vouch for places like Blue XTSea, Liquid Blue, Living Underwater, Tres Pelicanos, and others, and I doubt you'll go wrong with any of them.