Can a player move their king into check if the opponents piece that is able to take king, is putting opponents king in check?
It is never, ever legal to move into check. Period. No ifs, no buts. No exceptions whatsoever.
That's the best way to think about it, since it gives the least scope for confusion. If you want to think of it in terms of capturing the king ending the game, look at it this way: if you move into check, your opponent can capture your king, which immediately ends the game. It's irrelevant if your opponent's capture caused him to move into check too: you can't retaliate by capturing his king because the game already ended.
No. Check is check, even if the checking piece cannot legally move. There is extensive discussion of this in the answers to the question illegal (?) move can put a king in check?.
I'd rather be more cautious and answer this question taking into account the time control used in the game because, as you will see, there are some concessions that regulations grant when the players have less time to think.
If each player’s thinking time is at least 60 minutes:
The move in question is indeed illegal, as others have commented. End of story. Even though nobody notices the irregularity for a while, as the rules indicate (emphasis mine):
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.
End of story?
There might be a rare exception in case the game has ended (by checkmate or resignation) without the mutual check (or any other irregularity) being noticed, and the players have signed their scoresheets. I am not sure what happens if the illegal move is noticed after that because:
5.1.1 The game is won by the player who has checkmated his opponent’s king. This immediately ends the game, provided that the move producing the checkmate position was in accordance with Article 3 and Articles 4.2 – 4.7.
5.1.2 The game is won by the player whose opponent declares he resigns. This immediately ends the game.
When all the moves must be completed in a fixed time of more than 10 minutes but less than 60 minutes for each player; or the time allotted plus 60 times any increment is of more than 10 minutes but less than 60 minutes for each player.
In my opinion, the problem is more clearly described in other questions (both marked as a duplicate of this one):
Surprisingly, the rules specifically mention the situation of a mutual check:
A.4.4 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 an illegal position is still on the board, he shall declare the game drawn.
Otherwise, it is understood that the game can continue. For reference: A position is illegal when it cannot have been reached by any series of legal moves. (article 3.10.3)
Now, if this actually happens in a real game, the use of the word illegal to describe the move after the game has ended is arguable, because the move was illegal at its time, but now it is part of a legal game supported by the rules!
For other irregularities, the procedure is similar:
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.
To be completely precise, these Rapid rules do not apply if certain conditions are met. According to A.3.1, the Competition Rules shall apply to Rapid chess (i.e. the illegal move cannot stand) if:
A.3.1.1 one arbiter supervises at most three games and
A.3.1.2 each game is recorded by the arbiter or his assistant and, if possible, by electronic means.
Otherwise, the Rapid chess rules do apply and the illegal move has a chance to stand.
Finally, I have to mention that these Rapid chess rules also apply to Blitz, when all the moves must be completed in a fixed time of 10 minutes or less for each player (or the allotted time plus 60 times any increment is 10 minutes or less).