Where will you be staying ? If South of Caleta, the main marina located South of town (close to Hotel Presidente), you can expect to be picked up by any of the 80+ dive ops operating on the island. Less options quantity wise if staying in town (but still plenty of quality operations). If staying North of town, you may not have much of a choice unless you're willing to take a cab or drive to a pier.
Chances of being on a boat with only novice divers will depend on who else is booked as the same time as you are.
Provided you communicate clearly with your prospective dive op regarding your experience level and the type of diving you want to do, no good operation will take you on advanced dives such as Devil's Throat or Barracuda (strong currents).
Staying out of swimthroughs or above 60 feet however is a different issue, as most divers seem to want to do those. Which is why, no matter who you choose to dive with, I would recommand hiring a private DM just for yourself at least for the first day (approximate cost would be USD50 extra per 2-tank trip), so you can get a bit of experience with closer supervision. The following days you can just go with the group (all dives are lead by a DM and groups are kept to 8 divers max per Marine Park regulations) if you feel comfortable, or keep hiring a private DM so you can dive your own profile (for example if you don't want to go beyond 60 ft).
If you are really new to the sport, I'd recommand postponing the cenotes, or at least waiting until you do a couple of days of diving in the ocean (and hopefully a night dive too) and get the professional opinion of your dive guide before committing to diving the cenotes. They require good buoyancy and other skills as well as making sure you feel comfortable diving in a dark, overhead environment.
You can still plan for a trip to the mainland ahead of time, plenty to do there (snorkeling the cenotes if you realize you're not ready to dive them, visiting ruins, beautiful beaches, shopping, nightlife
), just don't pay in advance for a dive trip if it's non refundable.
The nightlife on the island is both low key and lots of fun. Cozumel attracts an "older" crowd of North American tourists, including lots of divers, who like to have a beer and relax and meet new people. Locals are really nice and enjoy chatting as well, in a respectful way. I found throughout my travels that it was sometimes hard to have a real connection with the inhabitants of the places I visited, mainly either because of the economical differences or the fact I was a blonde woman (bluntly said, felt the people I met were either not interested in talking at all, or did so only expecting a tip or a one night stand). Never felt that on the island.
My favorite thing to do at night in Cozumel, besides eating out and watching the sunset, is listening to live music in bars. Venues vary depending on the day. Good spots include Money Bar (South of town, near Fiesta Americana/Chankanaab) on the week-ends, La Hatch (in front of Villa Blanca hotel) and La Palapita next door, and Love Café Cozumel (on the beach just north of town). As an added bonus, those places are by the gorgeous ocean, great for sunset watching.
French Quarter (in town), as well as the above mentionned Love Café, contracts bands from other parts in Mexico. Ambar and Piratas (5 Avenida half a block south of the plaza) fill up with local youths when they have local bands playing. Viva Mexico on the waterfront in town also has live music starting at 11 pm Fridays (rock) and Saturdays (latin).
There are also places like Hard Rock Café, Carlos and Charlie and Señor Frogs, don't know what it's like there. On Sunday nights, the plaza in town is the place where everyone gathers for latin music and dancing.
There are also a few discos in town, but I've never been there, so can't comment.
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. We didn't stay out too late, but we enjoyed the Love Cafe and The Money Bar (great bands!). We also made a trip to the "Other" side, and loved Coconuts! We can't wait to go back, and I will not hesitate to use Blue XT again for our diving. 
