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The question is how the presence of suicides or freedrops in the input move is relayed to the post-move code so that it can check for their legality.

A freedrop will change the values of $origin, $dest, and $old. It will set $origin to false. A suicide is treated as a move to the null space. This is a space with an empty string for a coordinate. The important point is that it changes all the same values as a regular move does. So, doing a freedrop or a suicide on the second move is going to erase information about the first move. Promotions are special, because they do not erase information about the first move.

Is there a variable similar to 'dest' that tells which location they altered, or must I make a copy of the board before the move to compare to afterwards, to see if anything was changed, and what?

There is not. I suppose you could copy the board position, then check the hamming distance with the current position to see how many positions have changed.

Is the post-move code run after each move in the sequence, and then the post-game code at the end?

No, the Post-Move code is run after the completion of all the moves.

But then I don't understand how the code in the tutorial can work, because promotion happens only in the second move, so that the first move should be flagged as illegal when you run the post-move code after it, because it still has the Pawn on last rank.

As I just mentioned, it doesn't work that way. It plays all moves and fills in the variables with data on the last move. However, a promotion does not count as a full move. So, it does not change the data already filled in for the previous move.

The variables that get filled in after a move are most useful when you are not making a series of different moves. For your purposes, it would be better to check each move individually. To do this, store the position in the Pre-Move code, restore it in the Post-Move code, and then make an array of the moves by using explode on $thismove, and go through them one-by-one. This is what multi-move variants do.

But if you do it this way, you may have to code some things differently, such as promotion. You will also have the choice of evaluating a move before playing it or after playing it. Pick one and stick to it consistently.