-45 C was where I used to decide that I didn't really need a cigarette that badly.
Unlike most of the people who've replied so far, I have vaped at below -18, and I've never experienced anything like that. I usually just go out for a short period to vape, so my equipment starts off warm and the vaping keeps it from getting to cold. If I'm out for a while I keep the vape in an inside pocket so it doesn't get too cold. Now that I think about it, I don't think I've let my setup get down to really low temperatures so I don't know what the ejuice would do.
An even bigger concern is the effect of winter temperatures on batteries. I expect you already know that batteries don't work well when it's too cold, but recharging a cold lithium battery crosses the line from 'doesn't work well' into 'significant potential to cause an explosion'. You can actually end up electroplating the lithium in the battery if it's below zero, so always make sure your batteries have warmed up before charging.
Just as I don't understand how humans can continue to function at temperatures over 30 C (86 f), I've found a lot of people don't fully appreciate that life goes on at -30 C if you live somewhere that spends an appreciable amount of time there. People (and dogs) physiologically adjust to +/- 30 temperatures, so it's not as bad as it sounds to someone who isn't used to it. I doubt it would be wise to take a small dog on a long walk in that kind of weather, but the golden retriever I grew up with loved going for a romp at -18. We did have to watch out for ice buildup on her paws, though.