If you can keep the moisture out, BW is a good insulater. Perhaps some Christmas lights inside the BW will help on those really cold nights.
You can cut it down to whatever height you'd like, but if you want fruit, you need most of the P-stem. My DO flowered (and is now fruiting) after I cut about 12" off the top to take care of a little rot. mine went somewhat unprotected last winter with only a black lawn and leaf bag on it to keep out the rain. It didn't work...
