When thinking about if you can do some move, you look if your king will be attacked. This would lead to a recursion, but actually dont, since you it doenst matter if this enemy piece would be able to check/mate your king or if it would not be allowed to do the move because of check and mate rules.
The question is, its possible to on my turn put the enemy king into mate if this will put my king into check
No. Putting your king in check is not a legal move.
From this World Chess Federation handbook, section 3.9:
The king is said to be 'in check' if it is attacked by one or more of the opponent's pieces, even if such pieces are constrained from moving to that square because they would then leave or place their own king in check. No piece can be moved that will either expose the king of the same colour to check or leave that king in check.
Note that last sentence: no piece - which includes the king - can be moved that will expose your king to check.
No. You can't give check, nor mate, if your King is then under check. It's an illegal move.
No.
You can think about "white mating black" as a position that would inescapably capture black's king on the next move, therefore black has to surrender.
If white is putting their king into check in order to mate black, that means white will lose its king before it can capture black's king, so white would lose and that's why it's illegal.
