1. Horses can change lanes multiple times during their move. The rules don't say anything about this costing any movements points, so I assume this is free?
2. Horses cannot move diagonally. How are people handling changing lanes in the turns? It seems as though any lane change in either turn is going to be 'diagonal'.
1. No - you definitely must pay 1 movement point for each space you move through.
2. You simply move across a line into an adjacent space. As long as the two spcaes share a common line (border) they are orthogonally adjacent, and this is a legal move. If the two spaces only share a corner, they are only diagonally adjacent, and this sort of move is forbidden.