Honestly even snow isn't that bad. Bike tires, especially the ones on "hybrid" commuter bikes, cut through the snow pretty well. The real worry is that it freezes over and you can't see the ice.

In the coldest months I used to bike in full ski gear with rain pants rather than snow pants(easier to deal with around the gears but almost just as warm).

If it had just recently snowed or sleeted then I would take the full lane rather than be anywhere near the side of the road. Generally bike planes are entirely impassable once the plowing starts. That is where they dump all the snow so you will be taking the full lane anyway.