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I think I have a clearer idea of what you intend. However your page needs to be edited.

The Checkmate position: I see you meant to say that the Black King was in checkmate. I can see that it is attacked by the White Rook at the top level. I can see that the two White Queens above and below prevent it from escaping up or down. I cannot see the White Rook protecting either Queen, nor can I see any movement that the Black King could make to capture either White Queen. Nor do I see any White pieces attacking the eight adjacent squares next to the Black King on the same level. As far as I can see, the Black King could move to any adjacent square on the same level and escape check.

The Queen: People will not always assume that the Queen combines the movement of Bishop & Rook (it is after all a different game) and this should be explicitly stated. 

The Bishop: in 3-dimensional Chess the word "diagonal" has been used with various meanings. There are 26 directions of movement from a cell in cubical 3D chess:
- six orthogonal directions through the faces of the cube, changing only one co-ordinate with each step;
- twelve 2D-diagonal movements through the edges of the cube, changing two co-ordinates with each step;
- eight 3d-diagonal movements (sometimes called "triagonal") through the vertices (or corners) of the cube, changing all three co-ordinates with each step.
In order for the Bishop to not change colour, it would need to make the "triagonal" move when going from one level to the next, but make the 2D-diagonal move when on its own level. If this is so it should be explicitly stated. The Bishop in Raumschach and in many other 3D-chess variants moves differently, always using a 2D-diagonal move even when changing levels.

Your best description of movement is your description of the Knight-move.