use the original cube to get adjacency information (either 4 or 8 nodes depending on whether you want the diagonals connected)
Yes that does make sense, will have to sphere project the node data too in this case.
The only hassle is at the edges of the cube, not too difficult to cope with that though (and the corners have just 7 neighbours as well)
I think the issue is that each face of the cube is actually just a plane, so only contains vertex data about that face. I guess I could supply each face with there X+ X- Y+ Y- faces.