Your distances are pretty much the same as mine. And the winter temperatures you've mentioned are similar to the coldest winter temperatures I'll have to face. I don't have trouble getting around town without a car, even in winter. However, there will be some adjustments you'll make as you go car-free.
The cold isn't that big of a problem as long as you can dress for it. Check the winter cycling thread. Also go to a shop that sells outdoor gear and have them help you prepare for winter riding.
A potentially bigger problem will be balance and safety when you're riding on snowy or icy roads. I've found wide tires are a good idea. If you have a lot of ice, you can get studded tires.
It's a good idea to have a winter bike if you get snow and ice. The reason for this is the salt on the road can wreak havoc on a steel frame and the sand mixture is mean to your chain and drivetrain.
Also, make sure you're visible. Winter days are short and nights are long. You'll need a headlight, tail light and a reflective vest.
For the first year, give yourself permission to drive if necessary. Going car-free or car-light is a big adjustment and a huge learning curve, especially when it involves rough weather cycling. It's okay to go into it gradually.