Can I move my King here?

Then when the blak king moves is the white in check?

No, you can NOT move there... it may be pinned, but you would get taken first...

No, e4 is being attacked, by blacks knight, white can NOT move into check.

I have a problem confusing the d and e files. Also with the board not labeled you are making the assumption that a1 is in the bottom left. If for some reason we are viewing this from black's perspective, then e4 is then perfectly inhabitable (although the question becomes nonsense).

Fair enough, Bondiggity, fair enough...

(Also, nice profile picture)

If it's white move, wouldn't he simply take the knight, then try to work it for a black rook takes white rook, which is followed by white king takes black rook and stalemate?

NO U CAN NOT MOVE INTO CHECK

A pretty nice recovery...need to save some face, my dyslexic file syndrome is quite embarrassing :o

u cant move into check unless youre  game is stal mate

you can't move in check, and check is any position where unrestricted enemy piece action could lead to your king being captured on the next move.

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If you wonder why the 'unrestricted' is added, read the following:

The deepest objective of chess is: "capture the enemy king". This ends the game, and declares you the winner.

Putting this objective in the simplest possible rule (capture the enemy king when you can), and adding the second simplest rule (avoid leaving your king in a capturable position) would seem enough to make the rules reflect the objective of the game (next to the rules about piece movement).

However, the second rule seems to render the first one nonsensical because your opponent is unable to leave the king in a capturable position. Only the second rule exists on record.

The echo of the first rule, however, is found in the 'unrestricted' when defining check: capturing the enemy king is the primary objective of chess, and it does end the game. Therefore, capturing the enemy king can´t leave your king capturable: the capture is the last move of the game.

The reason that the board situation looks like a paradox is because the game doesn't seem to be about capturing the enemy king (it never happens), and because only the second rule is recorded.

The capture never happens because you would have to break the rule about avoiding capture. (when you can't but must move it is not a paradox: it's checkmate). The first rule isn't recorded because it it seems to add nothing over rule 2 and the checkmate clause (a mistake!).

The first unwritten rule, capture the enemy king is the number one rule of chess, and the soul of the game. the second written rule exists to reflect that, and can never alter it. Amen.

nope

FIDE Laws of chess. See 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."

Nope, you can't move yourself into check.