I don't have a ton of experience driving on ice and snow, so I'll let the experienced people continue to speak to that.
I have, however, been stranded for a day or so in an ice storm. Ended up holed up in a Panera Bread, and got to eat free soup all night, It could have been a lot worse though, 20 or so people died in Birmingham during that storm because they were stuck on the side of the road for a couple of days, and weren't prepared. The list of cold weather items and survival/prep stuff I keep in mine AND my wifes vehicle full time now includes:
-10 water bottles
-space blankets
-regular blanket
-tiny pellet stove
-tin cup for heating water
-coffee and tea
-4-6 mountain house dehydrated meals
-12 survival energy bars
-toilet paper
-complete change of clothes
-flashlight
-storm radio
-regular tool kit modified to fit the vehicle in question
-air compressor and tire plug kit
-all my offroad recovery gear stays on the roof rack of our 4runner all the time, so that's nice. My wife's fiat doesn't have any recovery gear, and I wouldn't really expect recovery gear to help it much if it got stuck. Its not much of a "go anywhere" vehicle.
Cold weather specific stuff (December through Feb for me, since we live in Texas with basically no winter)
-extra jacket (thicker than weather forecast would dictate
-coveralls
-extra socks, underwear
-extra shoes
-warm hat
If we're going remote, I'll up all that even further to basically include an entire set of sleeping bags, and most of our camping gear. The ultimate goal is me being able to say confidently that we could live out of our vehicle, even if its stuck in a ditch and not running, for multiple days, while staying warm and dry. Out of everything we carry, water is the biggest restriction, but I'm confident it would keep us alive for 2-3 days, and space is limited.
I'd suggest you think about all those items. We keep most of this stored where you can't even see it in the car. Personally, I keep my glock with me all the time too, and an AR in the car if we're going remote or if the storms are supposed to be really bad. People do stupid, crazy stuff when they're scared and uncomfortable. That's a different topic though, if you want to read into that, there are some long threads on firearms and overlanding in this forum.