It was the first day of running on snow and ice today. I can't say I was thrilled about this as I got ready but once I got started I enjoyed myself.
Between the music I was listening to and the intense focus on my footing, there wasn't any room in my head for thoughts of time, pace, or the like. Just noticing the sun on the snow and being grateful it wasn't colder.
I think the only way to approach running on Chicago sidewalks in the winter is to treat them as trail runs. You never know what kind of surface you'll be running on; it can switch from clear and dry to carefully packed down to icy knee-high caverns in the course of one block. It all depends on how the individual property owner maintains their stretch of sidewalk. So I step lightly and quickly and practice "monkey walk" when it looks icy (a technique I learned in kung fu years ago: keeping all joints slightly bent and sinking my center of gravity a bit lower).
And layer, layer, layer.