3.4.1) BrainKing does not allow me to make a move with my selected piece or the "swap dice" link is missing, but the target position (according to the dice value) is not blocked by opponent's pieces. Why cannot I make this move?

BrainKing always applied the following rule: If it is possible, both dice must be used. It means that some pieces can become "frozen" in certain positions because making a move with these pieces would create a situation where the second dice couldn't be used.