A regular (non-plugin) Prius will go maybe a mile on the battery alone, and only in the right circumstances. Acceleration on the battery is poor, and battery-only mode is limited to 40-45MPH. It'll start the engine for you on demand, so this is no problem when driving, of course, it just illustrates that the battery performance isn't all that important.
Depending on how you drive it, cold weather can hurt gas mileage. If you're driving in a way that the engine doesn't run much, then it may have to run the engine more to generate heat for the heater. However, if you're driving in a way that runs the engine a lot anyway (highway driving, decent amounts of acceleration hard enough to require the gas engine, etc.) then you get the same kind of "free" heat as a normal car.
Driving my Prius v, I'm battery-only basically when cruising on surface streets, sitting idle, or maneuvering around small, slow residential streets. And maybe, maybe if I'm accelerating away from a stop on a decent downhill slope with nobody behind me.