In Checkers/Draughts, can a King jump multiple pieces and end up where it started?
Nothing disallows this sequence of jumps. Pieces do not need to end on a different space to which they started the turn, kings are allowed to move and capture in any direction of play, and multiple jumps may be made if they are additional captures.

Further, if these were the only pieces on the board, or if no other pieces could have been captured off the square instead of continuing around the square: this move is not just legal, but mandatory.

If at any point there was an option to capture off the square, for example by continuing past the bottom corner to capture again in a straight line, the player moving the king may choose which capture to make, but they must make one of them.