Should I buy a red Z or a blue Z? Whether or not you drive the Z on snow/ice and what tires to use is pretty personal.
Do you know how to drive on snow and ice? If not, I'd leave it in the garage and get something a little heavier and with narrower tires to drive. From the few minutes of experience I have on snow/ice with my Z, it is NOT a Winter car - real light-weight and the wide tires don't help (kinda reminds me of the early, no-weight-in-the-rear Mustangs), but does OK if you slow down and stick to the rules of driving on slick roads.
If your roads are salted, you're going to have to give it a good bath frequently, especially underneath (as someone else mentioned, not a pleasant job in the cold). Corrosion will be your biggest concern if you know how to drive in Winter (and ppl that don't).
Personally, I try to keep mine in the garage when it snows; mainly because very few ppl around here know how to drive on it and I don't want to get hit. Since I live in S Arkansas, that's not but 2-3 days out of the year and the slick stuff seldom stays on the roads for more than a few hours in sunlight.
I have over 12 Winters experience with snow/ice (I'm no expert, but at least know that you don't lock up the brakes and turn the steering wheel to lock when you start sliding towards something heehee) and if I still lived in N Illinois I'd get some decent winter tires, hit the road, and take her to a heated car wash at least twice a week (or pay someone) to rinse off the salt.
YMMV