Even a light skiff of snow can turn an easy hill into an impassable obstacle, and deep snow can stop all but the most extreme 4x4s. Not only do you have the extra traction issues, but snow can hide rocks, water, mud, and deep holes. You should not venture into the mountains in winter conditions unless at least one vehicle in your group has a good winch, you have shovels, you are properly dressed, and you have supplies to spend the night.
The first rule of snow is that momentum is everything. As soon as your momentum is lost get off the gas immediately so you don't dig your tires in or create icy patches. Do everything slowly and gently and make sure to leave lots of room between vehicles.
Be very aware of the slope of the road. A slight slope for drainage that you didn't even notice in the summer can send you sliding towards the ditch in slippery conditions. You may need to steer slightly into the slope to keep moving the direction you want to travel.
Be sure to check the weather report for the area you are going to. Changing temperatures can drastically alter the driving conditions. It could be that the firm snow you drove on in the morning won't support your rig in the afternoon, or it could be that the slush you pushed through on the way in is now solid slippery ice.
Airing down for snow depends a lot on the conditions. If the snow is firm and your tires are biting in well then you should only air down enough to soften your tires slightly, typically around 20 psi or so. If you are trying to stay up on a crust or float on powder then air down as low as you can. I routinely run 8 psi in soft snow.
Running chains on all four tires will transform your rig in any kind of snow. If you only have one set then run them on the rear. Keep your tires pressure firm and remember that you have to keep your wheel speed to 30kph, regardless of what your actual speed is. Do NOT run chains if you are hitting rocks under the snow or you will destroy your chains.