You are doing a knight's tour, but this time, it's on the surface of a cube with chessboard faces.
If you move to a square on the same face you started, everything works just like on a normal chessboard. You can go to neighbour face by crossing the edge between the two faces, but move like both the starting and end face were on one plane. This is how you would get from A to B in the picture. You can also cross two edges, then the knight would travel from A, over the face B is on (while imagining A's and B's faces were in a plane) and then over to C (while imagining B's and C's faces were on the same plane. I hope I described it well enough so you can see how the knight still does its characteristic "2squares in one direction, 1 square perpendicular to that direction" move.
Note that if you only cross one edge, the color of the square does not change. Just ignore that and the color of the squares, this is all about hitting them exactly once.
1) Is it possible to do a knight's tour on a cube with chess faces?
2) If yes, what is the smallest cube you can do it on with edge length > 1?
3) What about open and closed tours?