Nothing horrible will happen to your bicycle in super cold conditions. Well ... I have heard of plastic parts snapping off, but other than that ...

Before it gets cold, you should wash your bicycle thoroughly ... clean off all the old summer greases, dirt, etc. etc. Then let it dry completely before going out in the cold. If you must add some grease, add something that is fairly thin and NOT WAXY. Believe it or not, WD40 seems to work quite well for me in cold temps ... and I've used a thin chainsaw lubricant too.

Although many people will disagree with me for some reason, and I'm not quite sure about the way to express this in physics terms, your tires will lose more air in the cold than in warmer temps. Try this experiment. Blow up a balloon in your nice warm living room. Fill it quite full so that it is definitely firm. Now take the balloon outside where it is cold and hang it from your mailbox or fence. Leave it there for a little while. Now go and take a look at it. You'll see that it has shrivelled so that it is not firm anymore. That's what can happen to your tires too. And when your tires are flatter, riding is more difficult.

However, if you are riding on snowy/icy conditions, you want your tires to be flatter ..... so you'll just have to live with the fact that riding the bicycle is going to be more difficult to ride.