I cycled through two winters here.
For snow and ice I used the bike with mud tires, they did great in the snow and ok on icy patches.
Layers and vented clothing so you can manage your overheating.
If you do get sweaty you can't stop for more than a traffic light.
Warm clothing for your legs too, keep the knees warm.
Learn to enjoy icicles in your beard. Prepare to be mocked when they're all on an angle from the wind.
The car drivers don't see you in the summer, why the **** would they see you in blowing snow, assume they can't. The drivers that do see you hate your guts, even if you're obeying the laws.
In those two years I got clipped by a mirror on a pickup truck on a snowy day, he sped off. And I got swiped by an asshole in a car on a bright sunny summer day, he took off and I followed him home. (Closed subdivision and the moron lived just a few blocks away. All I did was give him a piece of my mind and laughed at the scratches all down the side of his car.)
For the record, I stop at lights and stop signs and signal just as I would in a car, I've never been one of 'those' cyclists.
One more, your bike is going to be a rusty piece of crap if you don't keep cleaning and oiling the chain and other bits. The salt and road grime will destroy all that fast.
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