Most people are think that California is all sun and beach and are ignorant of the fact that there is a significant mountain range which runs the length of the state (including Mt. Whitney, the tallest peak in the lower 48 states at 14,505 feet). To get to the Gigafactory from Fremont, you have to drive over Donner Summit (7,026 ft.) and many Tesla employees routinely drive into the Sierras to ski. We regularly get 20+ feet of snow during the winter.
So, yes, the cars are designed for and regularly driven in cold and snow.
My S85D is absolutely the best snow car I have ever owned (better than the Audis, Land Rover, Subarus, Jeeps). It's like driving on rails. I've found it difficult to get the car to slip even by intentionally flooring it or slamming on the brakes on ice and snow.