If your insulation is damp, chances are good it could freeze or stay damp for a while. If not too much is involved, it would be a good idea to remove it to a warm place to dry out, or simply replace it. Never a good idea to leave wet material against ceiling drywall. It could dampen and ruin the drywall. If left too long, mold or mildew could result. The snow coming through the vents is probably an isolated event. Don't close the vent. Maybe just lay a sheet of 4 mil plastic over the insulation to keep snow off your insulation. Keep an eye on it. If it happens often, consider a better louvered/screened vent fixture that won't allow snow to blow in as easily.
What kind of insulation do you currently have in your attic? (Note: head on up there and check it out.) If it's batt insulation -- large units of fiberglass -- then that information will be printed on either the paper facing or the insulation itself. If it's loose fill insulation, which is far more common in attics, then you can usually figure out what the product is (pink, yellow, or white and very itchy = fiberglass, dirty grey = greenfiber, white and not very itchy = cellulose) and find it's value per inch on this handy table of R values.
For best performance, I'd go up to R-60. In a home built by your average builder in the past decade, R-30 meets code. Adding blown-in insulation, which you can do over fiberglass batts, is extremely cheap if you DIY ... although it's not a pleasant job to do. Rental of the machine from Home Depot or Lowe's is usually free with the purchase of a certain number of bales of insulation, and greenfiber especially is cheap. To bring my 1600 sq. ft. ranch up to R-60 would only cost me a couple hundred dollars in materials.
Do check at all ends of your attic. You might find that the areas close to the attic access are R-30 to meet code, but areas farther away might have less. I've known a few builders to be stingy with the insulation. One home I helped a friend with here in Houston had less than four inches of insulation in most of the attic.
One caution point is to make sure that in the process of adding insulation that any attic ventilation that you may have at the soffits is not blocked off. Soffit air is essential to the ventilation of your attic, if your home is designed that way. I would recommend installing Berger AccuVent baffles over every soffit vent before adding more insulation.
Last but not least, consider coating your roof with a reflective coating. You might need to get a contractor to do this, but it's simple and straightforward work. In the Inland Empire, nothing is more effective than keeping the solar rays from heating your house in the first place. Unfortunately, many find these roof coatings to be unsightly, so you might have to check with your HOA if you have one before applying anything.
Best Answer
You don't have to replace insulation unless you think that you can't dry it. Normally we would set up halogen lamps and fans to dry attics out. You can replace it too if that seems easier. How do you tell? Well after 1.5-2 days of lights and fans things should be bone dry. If not you probably need to replace. For instance in the winter we know we can't get enough heat and air going so we would just replace.
As for your ceiling, drywall can get wet and if it fully dries it retains 90%+ of its structural abilities. There was actually a study done I read about wetting and drying ceiling drywall 50+ times and basically they found if it dried right after it got wet you could basically get the drywall wet an unlimited amount of times. However if it is wet too long then the gypsum starts to break down. How do you tell? You will probably see all of you screws/nails popped out down.
If you just have a couple that popped out I would wait for it to dry then add screws back where needed. You might have to do some very minor mudding/sanding. If you don't want to do it pretty much any handyman/contractor/dry wall service can handle this - about the easiest job you can get.
If the drywall is just too wet to salvage, you simply need to remove all insulation on top of it, take it down and redrywall - and then put back insulation. This isn't hard but is a mess.
To get rid of the stains you need an oil based stain cover like the original Killz. If you try anything else you could end up putting 20 coats on. Usually 2 coats of Killz and then two coats of ceiling paint takes care of it - you can double coat within an hour for each with correct air flow.
As for your light - I would assume it is fine if it is dry and still working. I would not be concerned about that.