I have spent a couple of weeks in Niseko and Furano, both in Hokkaido over the past few years and have LOVED every minute of it and will be staying the season in Niseko this year. The amount of snow can go on the absurd some days! Although you have to watch, some resorts can have weird rules regarding off piste riding, so make sure you pay attention to ski patrol signs, as you can land in hot water if you break them. Niseko generally has pretty relaxed rules and has some awesome terrain to experience.
As for the traditional Japanese experience, you arent going to get too much of it up in Hokkaido as it is a pretty new area and a lot of westerners head there, so they have started to cater for that. Although if you have never been to Japan before it will seem pretty weird at first where ever you go. Regarding accommodation in Niseko or anywhere on the mountains look up google for 'pensions', they are bed and breakfast places and are usually pretty cheap and have a nice friendly atmosphere.
I could go on for pages about these places, but if you have any specific questions about Niseko or areas around there put them up here and i will try to help out