I had issues biking even in relatively mild winters. I actually had a Tyvek windbreaker. It seemed rather bizarre wearing what looked like a big water resistant envelope, but it worked. But I either wore tights or nylon leg warmers (basically just extensions of the bike shorts) and arm warmers, or even a long sleeve wool jersey. I'm guessing that doesn't work where it's cold enough that there's snow on the ground.
But for snow, I remember hearing what some had for the Iditabike. Stuff like dual tire setups (think truck tires) that would tend to float over snow (like snowshoes) rather than sink. That - and studded tires for dealing with ice or hard packed snow.