Changbaishan in Jan

1. Yes, it is possible to travel and see the scenery on the mountain. It will be extremely cold in Jan. so you'd better prepare enough warm clothes, boots and other packs. Generally, the view shall be clear to see all the scenic spots unless there is heavy snowstorm, when the whole scenic area may be temporarily colsed. Just pay close attention to the weather forecast in advance.
2. I know there is a Lanjing Days Inn near the North Slope gate, which has good service and pleasant facilities. Besides, I remember there is hotel in North Slope, near Changbaishan Waterfall, but forgot its name. It provides hot spring as well.
3. see answer no. 1.