Poor kitty! I would suggest following your vet's advice and get his teeth cleaned, although I would check around for a vet that will do it for a bit less. The first clinic I worked at (out in the country) did feline dentals for about 80 bucks or so, the one I work at now does them at about 100 or so. The procedure is relatively simple, but the anesthesia part is unavoidable. If you've ever had to hold your cat down to give him a pill, you'll understand why... Also, since teeth are such a vitally important part of a cat's anatomy, no one would want to risk damaging the root of the tooth or the gumline, hence another reason for putting them under. I would point a finger at the Calici virus, but I was told (just a few days ago, and now I'm glad that I asked!) that the virus is usually contracted early in a cat's life and comes from being unvaccinated. If you've had your kitty since kittenhood and he was (and still is) routinely vacc'ed, then you can probably rule out the virus. In any case, the virus causes the ulcerations (like timskitties said) on the gumlines and roof areas of the mouth. Generally, cats with the Calici virus lose their teeth at an earlier age than a cat who does not have the virus. It can also cause watery eyes, discharge from the eyes and nose, and sometimes sneezing or coughing/hacking. I think this virus can also weaken he immune system enough to worry about secondary illnesses, so it is good that you have Snickers on a preventative medication.
As for at home dental care, there is quite a bit that you can do yourself. Many pet supply stores and vet offices carry canine/feline toothscrubbers and specially formulated toothpaste. The vets that I work for say that pets' teeth should ideally be brushed once a day, but once a week is better than not brushing at all. Brushing your kittie's teeth will prevent him from having to undergo a dental again in the near future (say in about 6 months instead of a year or two after the first one). It will also give you the opportunity to monitor Snickers' mouth in case he should have a recurring disease of the mouth. Don't feel silly about brushing his teeth, either. Many of our clients seem to feel goofy just asking us, but it really is a good idea. When you get the test result back, go ahead and discuss at-home dental care with your vet. If you decide to go ahead and start doing the brushings, your vet or a tech should be more than happy to show you how to do so. Good luck and keep us informed!
P.S.- Just for additional info, if you've ever looked at the abbreviations for feline vaccinations, the "C" part of FVRCP is for Calici virus. The letters stand for different diseases, although in FVRCP, the first 3 stand for one virus. They are: Feline Viral Rhinotracheitis, Calicivirus, and Panleukopenia. The same goes for dogs with DHLPP, the letters stand for canine Distemper, Hepatitis, Leptospirosis, Parainfluenza and Parvovirus.