Hi Mike, and Welcome to the Great American RoadTrip Forum.
All my trips to North America have been between the beginning of March to the end of November, so I do not have extensive experience driving the roads in the winter. (Other than spending a few months driving with Jindabyne taxis during the ski season.) But I have spent quite some time in ID, and have a friend who lives there. Most of your trip is going to depend on checking the weather and road conditions at the time. It could almost be a case of taking it day by day, and with a month, you will probably be in a good condition to do that.
Some routes you may like to consider, weather and road conditions permitting. When leaving Boise, consider taking 55 north to New Meadows, where it meets up with 95. This is an incredibly scenic route... even in winter.
And to see more of spectacular central ID, consider 93 west from Idaho Falls to 75, and 75 south to Twin Falls (again weather and roads permitting). The latter will take you through Sun Valley Ski Resort at Ketchum and Hailey, and the wonderful Clarendon Hot Springs.
Not too sure about tent camping at that time of the year, in that area. From what friend tells and shows me, it gets pretty well snowed in sometimes. You will just have to take it as it comes, and be prepared to sit out a storm.
As for your other queries, I will leave it up to those experienced with those issues.
Lifey in Melbourne