Growing up as a teenager in Montana, bikes were an important part of transportation - at least when the weather was good. In our neighborhood one of the kids had an old Schwinn Stingray with very wide tires and more than a bit of rust. He didn't care if it was left out in the snow and the tires were great for traction. In normal weather it was terrible bike, but in the snow it was enormous fun. The tires were just big enough to grip all but the lightest powder and it was remarkable to be able to cruise around like that. I'm sure we were burning an enormous amount of calories in the process. I remember being out in single digit temperatures in a light coat and getting nice and toasty.
A certain type of wide tired bike has evolved that has *really* wide tires - anywhere from 3 to 3.8". These are usually made for riding in the dirt and mud, but they'd make incredible snow bikes. Here is one from Surley - if you want a different take on Winter sports and don't have to fight sliding cars, this might be great fun. It would probably be amazing exercise, so I'd put a hot chocolate holder on it.