When is a move "completed"?

 

Yes, of course.

After a player has released a piece on destination square, by touch-move rule the move is final... unless it was illegal. Correct?

The move is completed when you hit your clock.

FIDE listed it as first, but NOT the only way to complete a move.

"A player must be allowed to stop his clock after making his move, even after the opponent has made his next move. The time between making the move on the chessboard and pressing the clock is regarded as part of the time allotted to the player.

So: if a player releases a piece at destination, but then fails to hit the clock, the opponent is required to permit him to hit the clock when he finally remembers to do so, but is not required to hit his clock himself, nor wait for the player to finally remember to hit the clock - the opponent is allowed to take the opportunity to reply in the player´s time.

Correct?

FIDE articles 3 & 4 deal with moving the pieces. The term "completing a move" does not feature at all

Article 6 deals with the clock and this is where the term "completes a move" is introduced

It is quite clear from 6.2.a.2 that if a player fails to press their clock to "complete" a move, then the move is deemed to be "completed" after they make their next move.

This rule reinforces articles 3&4 in showing that any legal move is considered "made" once the player has released the piece and must stand, irrespective of whether he presses the clock or not.

The rule as written, would allow an entire game to be played all on one players time, although I am sure an arbiter would have words with the players

Then what are the possible outcomes if a player makes an illegal move but does not complete it?

If a player makes an illegal move and does press clock on it, the opponent and arbiter are both entitled to claim illegal move, which would result in the opponent gaining time and first count of two towards forfeiting the game.

But if a player makes an illegal move and does not press clock?

If he can move the piece that was illegally moved, he must move it. Otherwise, he has to make a legal move.

So what should be done if an opponent has replied to an illegal move?

If a move is a reply to an illegal move, is the legality of the reply move defined by reference to the (illegally reached) position on board, or otherwise?

The rules on yhe FIDE website are pretty clear.

 

7.5 

If during a game it is found that an illegal move has been completed, the position immediately before the irregularity shall be reinstated. If the position immediately before the irregularity cannot be determined, the game shall continue from the last identifiable position prior to the irregularity. Articles 4.3 and 4.7 apply to the move replacing the illegal move. The game shall then continue from this reinstated position. 

If the player has moved a pawn to the furthest distant rank, pressed the clock, but not replaced the pawn with a new piece, the move is illegal. The pawn shall be replaced by a queen of the same colour as the pawn.

After the action taken under Article 7.5.a, for the first completed illegal move by a player the arbiter shall give two minutes extra time to his opponent; for the second completed illegal move by the same player the arbiter shall declare the game lost by this player. However, the game is drawn if the position is such that the opponent cannot checkmate the player’s king by any possible series of legal moves.

So, that's the count of illegal moves completed by each player.

If, after an irregularity has been noticed and on the rereading of scoresheet, both players are found to have completed two or more illegal moves after the first illegal move, what's the result? Draw? Or loss by the player whose second illegal move came first?

That is a little more subjective but my guess is if multiple illegal moves happened, the arbiter would count that as a first offense by both players. The next illegal move would result in a loss.

There are specific rules for ill supervised rapid chess:

A4b: "An illegal move is completed once the player has pressed his clock. If the arbiter observes this he shall declare the game lost by the player, provided the opponent has not made his next move. If the arbiter does not intervene, the opponent is entitled to claim a win, provided the opponent has not made his next move."..."If the opponent does not claim and the arbiter does not intervene, the illegal move shall stand and the game shall continue. Once the opponent has made his next move, an illegal move cannot be corrected unless this is agreed by the players without intervention of the arbiter."

and A4d: "If the arbiter observes both kings are in check, or a pawn on the rank furthest from its starting position, he shall wait until the next move is completed. Then, if the illegal position is still on the board, he shall declare the game drawn."

So... In ill supervised rapid chess, "making" a reply to an illegal move, even if the reply move is legal, forfeits the opponent´s right to claim a win. It also bars the arbiter from further interference, even if not only the move but the position is illegal and the fact is obviously provable, unless the illegality of the position is of the two specific types of both kings in check, or pawn on last rank.

In slow chess, or adequately supervised rapid chess, both players and arbiter keep the right to call any illegal move indefinitely.