You know your finances best, but it CAN be done pretty cheap.

What kind of bike do you have right now? Have you tried riding it in the winter? You can get a feel for winter riding without committing to commuting just by taking it out for an occasional recreational spin on local sidewalks / quiet roads / trails.

A rigid mountain bike is a good choice. I don't think going single speed is a real advantage: it's nice to have the option of gears. If you use friction shifting instead of indexed it usually works pretty well. If your shifting freezes up...you temporarily have a single speed bike, anyway. Or maybe a three speed, as my front derailleur has almost never stopped working.

I like nice, knobby "standard" sized mountain bike tires(26" X approx. 2.0"). I like the knobby ones because you can get a bit of a paddlewheel action going, as opposed to smoother tires that tend to spin out more. The DIY studded tires work pretty well, and are cheap to make.

The only advantage I've found of drop bars for winter riding is that you can more easily tuck down into an aerodynamic position when riding directly head-on into a stiff, freezing wind. That can be nice when the wind chill is -30C plus whatever you're generating by riding. My current winter bike doesn't have them, though, and I don't miss them.