I didn't have problems with the traffic, I used secondary roads, but in main roads connecting big cities you'll find heavy traffic. anyway, usually you have a big shoulder for bikes, specially near cities, even in most tunels (lots of them in japan).
I slept in different places, always depending on the "moment", you wont have problem with free camping (I slept in small parks like rest stops by the road, even without tent, but then the problem were the mosquitos, so then I should go into the tent which was very hot!!, but there's not many campgrounds, actually I never was in one of them. You will have visit of monkeys at night
sometimes, specially in Hokkaido. sometimes I slept in Min-shu-ku, its like a Guest houses in rural/turistic areas (with onsen...) and too expensive (cheaper than ryokans)
hope good recuperation from injuries (I have some shoulder problems too
), I wasn't in Fuji, basically for the same reason, I think it's a mountain to enjoy in winter, YES!!!