I wear normal shoes. Unless you are running in deep snow or something, the only issue is when you get your feet wet by stepping in a puddle. The only real time I stay inside is when there is lots of black ice out on the sidewalks, because that's just asking for an injury. I've never used them, but I've heard good things about Yak Trax so you may want to check them out.
Otherwise, the clothes you need are way less for running than cycling. At 32F (0C) I'm generally still wearing shorts, long sleeve shirt, light-ish pull over, gloves, hat. Below that much more and I start throwing on tights and other upper body layers.
In the winter I generally slow it down to be safe, and hit the trails where things aren't as slippery (plus more wind protection).