with your dive operator and DM. They will know what will be the best places to dive for the conditions (time of year, currents etc). If they know in advance about your requirements they can prepare. 30 minutes out on the water is the wrong time to try to tell them that you want a shallow dive (or whatever).
There are plenty of great things to see in Cozumel without having to go deep.
Only you can know whether your wife's diving enjoyment would be greater if she is diving with you or without you. I would think that she would want to have you around for at least some of the dives, especially the stressful first ones. After that, it sould be up to the two of you. Personally I like the shared experiences the most. Common ground and all that.
I went with a group, but when they decided to take a day off, I dove with a small group arranged by the DO and we had a great time because they matched us by skill level and (diving) desire.
There are places at Palancar that are deep without being scary, and I would recommend that reef very highly. It is so large that there are at least 3 separately defined areas of the reef.
2.) As for food:
I agree with so many people that Guido's (on the main waterfront road Melgar) is one place that you must try once. The Lasagna was great. I forget what I had the other time I ate there, but everyone loved whatever they ordered there.
Ditto on the La Choza recommendations.
For authentic and inexpensive lunches, try Jessy's next to La Choza and definitely try Comida Casera Tonita a block or so further up the road than La Choza. Extremely reasonably priced authentic food, daily choices. The place is very popular with the locals, and they know value.
Prima and La Mission (3 of them) are notorious tourist restaurants. Usually good recommendations from people that have eaten there, but on the expensive side.
When we wanted a break from Mexican food, we went to TGI Fridays and ate like the American pigs that we are.
Those are the main ones that I can vouch for.
Have a great time, and let us know how it was when you return.
Wristshot