I give the below average rating because your description does not give me enough information to play the game. I make the following points:

A) The opening scenario is unnecessary and inappropriate. A force of commandos & terrorists would not bring their King into a battle, nor would the forces be identical. Chess dates from the time when Kings actually rode out to battle. A besieged castle would be a better description.

B) The 3-D board has all the white squares above or below the other white squares in a vertical column; the black squares are similarly arranged. This means that a Rook moving up or down (like a lift in a multi-storey building) would stay in cells of the same colour. Thus if you were looking at the building from the front, you would not see a checkered board; instead you would see four vertical columns of white squares alternating with four vertical columns of black squares. Was this your intention?

C) This being so, I do not understand how the Bishop moves when changing levels. In order to not change colour it would have to move "triagonally" (that is, through the corner of the cells), not diagonally. It would have to follow the same path you illustrate for pawn captures when they are going up or down. An illustration of the Bishop's move would help.

D) From your description, the Queen's move is identical to the Rook's move, namely an orthogonal move in all six directions. The only difference is that the Queen has the additional power to place an enemy King in check if they are both on the same level with no other enemy piece on the same level. Was this your intention?

E) Going from your description the checkmate position you illustrate is not checkmate at all. I presume that the Black Queen is the one under the White  King. The Black Rook does not protect the Black Queen as it is in a diagonal line from the Queen. The White King could move to any of the five squares surrounding it on the same level without being threatened by the Black Queen or by any other piece. It could also move upwards diagonally or  "triagonally" and not be threatened. The only square that it could not move to on the level above it would be the square directly above itself. In total I see eleven cells to which the King could move to escape check.

Perhaps illustration of the moves of the Rook, Bishop & Queen would assist, as I can only go by your verbal description.

The game will take quite a long time to play. This is not necessarily a fault. However it was for this reason that Ferdinand Maack created the original 3D chess as a 5 x 5 x 5 game.