Winter riding: Is it safe to ride on a BSO/cheap bike at temperatures below -15°C on icy or damp roads?
I can only speak from my experience, as a winter rider in Norway (around the capital).

When the winter sets in I usually just end up putting more clothes on myself, and studded tires on my bike. I avoid hard biking (biking so that I get tired, breathing heavy) when it's below minus 10/15 degrees C, because breathing in such cold air can be damaging for your lungs.

Of course, with all the extra salt (at least in Norway) the bike parts get worn out faster, and I usually end up changing at least the chain after the winter, and maybe also the cassette. I have some friends that usually buy used cheap MTB and throw them away or sell them at the end of the winter season.

Edit: One other thing that is important, is to be visible. Depending on the road you're riding, you may share it with other cars or cyclist/runners/walkers. In Norway it is mandatory to have light (yellow in front, and red in back), plus a bell. In the winter I also use a safety vest (they may look silly, but it's not cool to be hit by a car!), plus a stronger light in front, than in the summer. Just be aware that the light should point down, so you don't blind other people.

I think the best solution is just to try it out, and take it easy. Wider studded tires will help you.

Also, be careful about sharp turns. They are dangerous in the snow/ice/winter.