I live in a very cold part of the world. 35 below zero is not uncommon December through February. There isn't much about the cold itself that will harm your bass. You might have trouble with some finishes, but probably not. You're more likely to have trouble with the plastic parts on your amp, your speaker cab, your bass bag, etc. A lot of plastics are useless at those temperatures -- they get brittle and break.
With both temp and humidity -- but more with humidity -- what you need to watch for are rapid changes, moving quickly from very cold to very hot, very wet to very dry. With a woodstove in a small cold cabin, you may indeed run into some of these issues. I'd say: keep the bass far away from that stove, especially while the cabin is warming up from dead cold. You might think about a heavily-insulated bag or something, just to attenuate that warming/cooling process, to slow it down a bit.
