Can I stain wood in sub-optimal temperatures?
I haven't seen anyone test 'how do bad conditions affect staining?' Usually what you'll see is someone posts a picture of a bad stain job and they ask 'what did I do wrong?' The answer is typically temperature, humidity, or technique. 

Stains work by penetrating into the wood, getting absorbed into the pores of the wood. Anything else that gets into those pores will hinder absorption, and you'll end up with a blotchy inconsistent stain. That's why you want your wood dry.

Low temperature is less important than moisture, with the exception that if there is frozen water in your wood, that'll mess up the stain as well. If the wood is cold but stays dry, you should be ok. 

In ideal temps, the stain should dry in 1-2 days. Low temps will slow the drying process, and if snow/rain gets on the wood before it's done drying, you're in trouble.

If you can keep the wood protected from rain while it dries, you should be fine.