Does it matter whose move it is? For instance if a given arrangement of pieces appears once when it is white's move, and then later it appears again when it is black's move, do both of those instances count towards the three-fold repetition?
Yes it does matter. The position is only repeated when the board position is the same and the same side has the move, otherwise it is a new position entirely.
Also castling rights, and en passant options must be the same for two board positions to be identical. For example white's uncastled king who has the option to castle, moves e1-e2, and then next move returns to e1. This is not considered the same position as what existed before he moved e1-e2 as he now has lost the right to castle.
Does it matter whose move it is? For instance if a given arrangement of pieces appears once when it is white's move, and then later it appears again when it is black's move, do both of those instances count towards the three-fold repetition?