Is this an alternator problem?

Extreme cold weather, which you endured, causes the coating on the wires to stiffen and a crack can be created. This sounds more like the case and a short was created. I would immediately take it to an electrical specialist for an evaluation.

It is doubtful to be an alternator problem, as they do not come-and-go. The alternator usually works or not.

FYI
Today I took it in and they replaced the front engine seal (which was spewing oil under the hood, thus the burning smell). My car was also losing oil rapidly, according to the repairers. I asked how much $ it would be to wait for just the seal itself and they said about 5 business days. They said my car would lose power by then. So they had to order the whole engine seal 'kit' in order to get same day delivery/installation. Total damages in Canadian dollars: $286.63
Opinions?

I am confused, are they replacing the seal as well, or is that just the price for the kit? If that includes labor, that is a pretty fair price--I am assuming this is the crankshaft seal. There is a good deal of work to replace it.

Oh, I'm sorry. In my previous post, I meant to say: "I asked how much time it would take to just wait for the seal itself"
Yes, they did replace the crankshaft seal with a whole new kit. And $286.63 is including labor. Thanks so much for your input, much appreciated!

Glad you got that issue resolved. Let me know if there is anything else I can ever help you with. Thanks!