Relations between Japan and the West explained

Why are netsuke (carved kimono toggles) so famous in the West as a representative Japanese art and yet so little known or seen in Japan?

For one thing, they were just seen as a craft not an art form in Japan and so were usually ignored. They probably became popular in the west because they could be picked up cheap and easily transported due to being small and difficult to break. In fact, a lot of the netsuke in the Victoria and Albert museum etc. were specifically produced for export.

Why are Western museums full of ukiyoe Edo era woodblock prints and Japanese ones just full of byobu folding screens etc?

Again, ukiyo-e was not considered art by the Japanese of the time. It could therefore be picked up very cheaply. In fact, the story is that the first ukiyo-e prints to be collected in Europe arrived as packing for pottery and were meant to be thrown away.

Why is shunga (sexual ukiyo-e Edo period woodblock prints) so famous in the West but so little seen in Japan?

They are so sexually explicit that it is possibly illegal to display them in Japan, although people have occasionally got away with small exhibitions of them.