Depending on your fitness level, 10 miles might be a stretch at first, depending on the uphill grade but within a month it will be no problem. Depends on how presentable you need to be/whether there are showers at your work or at a nearby (cheap) gym. You WILL sweat going up a grade. Otherwise, find a way to transit in but bike home.
There's quite a few threads on here about biking advice, but some basics:
-Hybrids rule. I love my Trek FX I picked up used after advice here. Even the stock tires have great grip in snow/slush. If we had more ice storms, I'd get studded tires, but it rides incredibly well - better than my old MTB.
-Resist the temptation to buy biking clothes for winter before experimenting. If you must buy something, Sierra Trading Post (with their regular e-mail 30% off sales) can't be beat for base layers, rain jackets, wool socks, etc.