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It depends entirely on the move notation you start from.

It uses simple algebraic notation that always includes both coordinates but doesn't necessarily include the piece, and all it needs for castling is the move that would otherwise be illegal for one of the two castling pieces. So, castling gets done with only one written move instead of two. When the Rook initiates castling, this is easily spotted, because it moves to an occupied space or hops over an occupied space, and that occupied space is where the King is. But when the King initiates castling by moving two or more spaces, this could also be a Rook or Queen move. If the player used piece notation with the move, there wouldn't be any problem, but some players left out the piece notation in the move, because Game Courier didn't require it to know what to move.

I don't think the problem can be solved in general. The first move of a Knight or Bishop could come from two locations on the back rank. And if both these locations are visited by an enemy piece before the occupant of the other moved (or the game finished before the other moved), you would never know which of the two it was.

No, there are other methods for identifying pieces besides noting how the piece moved. Pieces are associated with binary numbers corresponding to which potential pieces they might be. When all pieces of a particular type have been identified, the bit for possibly being that piece will be turned off for all the other pieces. By this means, process of elimination is used to identify pieces that never moved. Also, the two Bishops are distinguished from each other, as they go on different colors, and since Black's pieces mirror White's, a move by a piece on either side is enough to identify it for both sides.