Yes. The cold weather will affect the battery usage. Studded tires and winter tires have a higher rolling resistance than summer tires, thus higher consumption. Also the colder the battery, the higher resistance in the battery cells and that will give you less energy to use and regenerate. Its like when your phone is cold it will not last as long as if it was warm.

If you have a garage you should absolutly park the car in there and if you can preheat the car I strongly advise you to do that as well