Can a win be claimed on the second illegal move if the first wasn't raised? First of all, I don't think this is a duplicate of what happens on the first illegal move in a game. This is what happens after the second if the first is unclaimed.

The setting is a tournament under FIDE rules (2018).

Can a win be claimed after the second completed illegal move completed by one side, if neither player nor the arbiter intervened upon the completion of the first?

Assume the arbiter was summoned by the claiming player, and sees the game and its score for the first time. Assume that both players recorded the moves faithfully, illegalities included, such that they agree on the history of events and that there is no doubt that more than one illegal move was completed. (See below for an example.)

The relevant articles of the FIDE Laws of Chess seem to be:


  7.5.1     An illegal move is completed once the player has pressed his clock. 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.


And


  7.5.5     After the action taken under Article 7.5.1, 7.5.2, 7.5.3 or 7.5.4 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.


As a concrete example, look at the following game and assume the arbiter was summoned after the move 24. 0-0-0. Both players recorded the game and agree the game score is correct.

[FEN "rnbqkbnr/pppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR w KQkq - 0 1 "]
1. e4 e5 2. Nf3 Nc6 3. Bb5 Nf6 4. Nc3 Bb4 5. Bxc6 bxc6 6. d4 exd4 7. Nxd4 Ba6 8. O-O d5 9. exd5 cxd5 10. Re1+ Ne4 11. Qg4 Qf6 12. Bg5 Qg6 13. Qf4 O-O-O 14. Bxd8 Rxd8 15. f3 Nxc3 16. bxc3 Bxc3 17. Reb1 Bxa1 18. Rxa1 Kd7 19. Qf5+ Ke8 20. Qxg6 hxg6 21. Nc6 Ra8 22. Kf2 O-O-O 23. Ke1 Re8+ 24. O-O-O


(Aside: this is probably a plausible game between weak players, so the question has maybe more than academic value.)

The illegal moves are 8.0-0, 13...0-0-0, 22...0-0-0 and 24.0-0-0. (Relevant articles of the Laws: 3.8.2.2.1 for the first three, 3.9.1 for the last.)


Black attempts to claim a win under 7.5.5. Does this stand?
Could white have claimed a win had she summoned the arbiter after 22...0-0-0?
Extension: if the claimed win(s) does (do) not stand, under 7.5.1, from which position does the game resume?
Black attempts to claim a win under 7.5.5. Does this stand?


No. According to 7.5.5 -


  After the action taken under Article 7.5.1, 7.5.2, 7.5.3 or 7.5.4
  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.


After the first illegal move the arbiter gives the opponent an extra 2 minutes. He will also explain his decision to the two players and warn the player that a second illegal move will lose the game. Punishment for a second illegal move can only occur if the punishment for the first illegal move has been made. 

There are recent judgements (for which I don't have reference to hand, but I suspect came from one of the FIDE Arbiters Magazines) saying that only one illegal move can be punished at a time. For instance, if a player promotes illegally by not replacing the pawn while still in check, he can not be punished for two illegal moves.

Therefore after the first reported illegal move only the punishment for a first illegal move can be applied. Another illegal move must occur for the punishment for a second illegal move to be imposed.


  Could white have claimed a win had he summoned the arbiter after
  22...0-0-0?


No, the same arguments apply.


  Extension: if the claimed win(s) does (do) not stand, under 7.5.1,
  from which position does the game resume?


From the position before the first illegal move. The punishment and reinstatement of the position are unrelated. Article 7.2.1 is relevant here -


  7.2.1 If during a game it is found that the initial position of the pieces was incorrect, the game shall be cancelled and a new game shall
  be played.


Regardless of fault or punishment the entire game should be played according to the FIDE Laws of chess.

Note, I had a very similar problem arbiting a tournament in 2014 when a player came to me to report a third illegal move by his opponent and claim the game. Since it was the first illegal move he reported I treated it as the first illegal move. Since the previous illegal moves had involved his opponent not getting out of check and the position corrected by the players at the time there was no need to go back beyond the current move.

PS Before tieing oneself into gender-anguished knots it is worth noting this line in the introduction to the FIDE Laws of Chess -


  In these Laws the words ‘he’, ‘him’, and ‘his’ shall be considered to
  include ‘she’ and ‘her’.