So if your opponent promotes a pawn to a Queen and accidentally places
your Queen on the board instead of his own, can you claim a win
because of an illegal move?
No, you can't. There is no illegal move until the player either presses the clock or makes another move (after the opponent has moved).
Here is what the FIDE Laws of Chess have to say:
188.8.131.52 When a player, having the move, plays a pawn to the rank furthest from its starting position, he must exchange that pawn as
part of the same move for a new queen, rook, bishop or knight of the
same colour on the intended square of arrival.
3.10.2 A move is illegal when it fails to meet the relevant requirements of Articles 3.1 – 3.9
6.2.1 During the game each player, having made his move on the chessboard, shall stop his own clock and start his opponent’s clock
(that is to say, he shall press his clock). This “completes” the
move. A move is also completed if:
184.108.40.206 the move ends the game (see Articles 5.1.1, 5.2.1, 5.2.2, 9.6.1 and 9.6.2), or
220.127.116.11 the player has made his next move, when his previous move was not completed.
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.
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.
So, first of all, yes, promoting to the wrong piece is an illegal move (18.104.22.168 and 3.10.2)
Secondly, just making an illegal move does not attract any penalty (other than having to make a legal move with the piece touched or captured). The move only attracts punishment when it is completed (7.5.1 and 7.5.5) by pressing the clock or making a subsequent move.
Thirdly, you cannot claim a win for the first completed illegal move, only for the second (7.5.5). This rule now also applies to rapid and blitz.
And if you don't claim, but continue to play on without correcting the
situation, is that Queen officially yours?
At standard time controls, no. According to 7.5.1 the position before the illegal move must be reinstated and a legal move made.
However, rapid and blitz are different.
Appendix A. Rapid chess
A.4.2 If the arbiter observes an action taken under Article 7.5.1,
7.5.2, 7.5.3 or 7.5.4, he shall act according to Article 7.5.5, provided the opponent has not made his next move. If the arbiter does
not intervene, the opponent is entitled to claim, 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.
Appendix B. Blitz
B.4 Otherwise, play shall be governed by the Rapid chess Laws as in
Article A.2 and A.4.
In rapid and blitz the standard rules for punishment only apply if there are conditions which will only occur in very high level tournaments, if then. These are that the moves are recorded independently and there are enough arbiters (one per 3 games at rapid and one per game at blitz) to properly see what is going on. Otherwise article A.4 applies and you can keep your extra queen.