No problems with cold
I have a 2nd generation iPod Nano I purchased in 2007. It has been a permanent fixture in my car since and is still going well enough (*see below). I live in VT where temperatures will frequently drop near or below -10 F.
Odds are my Nano has a NiMH battery, not a Li-ion like more modern devices use. Li-ion batteries reportedly do not charge well in extremely cold or hot temperatures. In particular, charging while the battery is hot can shorten its life significantly.
On the rare hot day in VT (90 F or higher) when the car has been sitting in a parking lot, my Nano will refuse to turn on. But on those days I just unplug it and jam it in the AC grill for a minute or two and it'll start working just fine.
After 7 years of this abuse, my Nano does frequently require a manual reset when I get in to start the car, which is annoying. I'm not sure if this is because the battery is just getting old or if it's had some permanent damage due to the harsh temperatures.
As another poster mentioned, I've never had a problem with the LCD screen at subzero temperatures other than it can be a bit slow to display. That's not unusual and is typical for any LCDs included in cars by the manufacturer, like radios. Computer electronics love cold temperatures, so that's not a problem. The only concern would be the battery.