But you actually only play the game in two dimensions, because the pieces cannot move up or down. Normal chess is a game happening on a 2D plane whether or not the representation of the board and pieces is 2D or 3D.
No, because that again depends on the representation. You can store the information of moves and chess position in 1D. As for example, two players can play a game of chess just by exchanging moves via morse code.
