The Flying Knight

From the wikipedia article on monster chess, as this is obviously a variant of monster chess (one knight instead of four pawns):

"The white king can move into check on the first move of the turn and move out of check during the second move."(The article has white as the player moving first)

http://en.wikipedia.org/wiki/Monster_chess

OK, so black can approach white just like in normal Chess (1 empty square between them), and then white is in check, (because when he would pass his turn, the black King could now capture the white King), so he has to increase the distance again to two empty squares between them. And when he cannot, because he is driven against the edge, white is checkmated by the black King alone.

In the diagram I gave, could black just step his King to b6?

The black king, because he does not jump, can't go through check.
I already said this. If the diagram you posted is the one with the two rooks, then the answer is no.

The black king doesn't need to jump. The king just needs to be out of check by the end of the turn.

True, you already said this. But you are constantly saying things that contradict each other. For one, 'jumping' is moving over an occupied square. Not moving over a square that is attacked.

If the black King is not allowed to pass through check (i.e. be subject to capture in the hypothetical case that black would skip doing his second move), then there is no reason why the Kings should stay further apart than in normal Chess. One empty square between them would be enough. White would not check black then, because he can only do one step. And black would not check white, as he cannot step next to the white King with his first move (as this would be passing through check), and thus can never capture the white King with your King's second move.

Note that not passing through check is an extra rule, in deviation from FIDE rules, which in general only care about the position at the end of the turn.

It is impossible for the side with two to lose if they go first. It gets two moves to white's one, and a win is essentially guaranteed. 

As an example: for algebraic notation I will demonstrate white's two moves as two seperate moves, with the final move being the piece's final placement, and the third move as blacks. Assume black has all pieces and white's king and knight are on starting spots.

1. Nd2 Nd4 e5

2. Nf5 Kd1#

In two moves white can take black's king, no matter where he moves.

The way black could avoid this would be for his first move to guard the f5 square, but with pressure from the king in a few turns it would be inevitable.

I guess this is why black gets the double move, and not white.

The critical issue remains which rules one adopts to restrict the first move of a black move pair. In Monster Chess there are no extra restrictions: this means the black King will be allowed to step to squares attacked by enemy pieces, as it cannot be captured there (because it is not yet white's turn), as long as he ends up unchecked after the second move. This in particular means black can step his King next to the white King, as long as he then steps on. As he can capture the white King from there with his second step, this means white is already in check when the black King is two steps away from him, but that the black King is perfectly safe there, and allowed to deliver such checks. As a consequence, K-vs-K end-games are easily won in Monster Chess.

If the additional rule is adopted that the first black move must be legal under the fiction that it was the only black move, (i.e. 'no passing through check'), then the black King cannot check the white King from a distance of two steps, as he is not allowed to make the first of these steps. Of course the white King cannot capture the black King from this distance either, as it is limited to a single step in the first place. So Kings can approach to the same distance as in FIDE, and neither of them would be in check.

It seems that what the OP describes is neither of this. Which means there is need for yet an additional rule in addition to FIDE and the 'no passing through check' rule.

Black wins easily.  This is a scam to try to win money off of people who think white should have the advantage, but white falls quickly.

Essentially, as I'm understanding, the Black King is upgraded to the powers of the Lion (from Chu Shogi), where it can make two King moves in one turn; and, furthermore, where it can make a capture on a piece adjacent to it only to retreat to any available safe square on its second move. It's nothing new: we all know that a good avoider is placing a piece adjacent to Black's King at all times; and placing a piece where a black King would get there.

How exactly is victory achieved by White, since he/she is up against 2 enemy Kings? Do we do it Ace569er style, where we capture Kings until there's only one left, in which the last King is to be checkmated? Or, do both Kings have to be checkmated at once?

Also, what exactly is the significance of the Knight? If it has already been addressed in the mountain pile which are the posts before me, it would be nice if someone can point it out to me; since I don't have the time to read everything tonight.

While, again, I haven't read the entire thread, here is my recent discovery:

Black may have an area attack of a 5x5 square, but the White King will not be affected by it; only by the default 3x3. Since a Black King, regardless of its double attack, can still never be allowed to be adjacent to the White King by the end of its turn.

In fact, if one wants to see it the following way, White can  be place adjacent to one of Black's King, due to Black's 2-move nature, in which moment Black must move its adjacent King by the end of his/her first move. The idea is that the verdict of King checking King would be determined upon Black's 2nd move, since he/she is still allowed to make one more move after the normal 1st; such which would give leway to White's King, because White is not technically in check with Black's King prior to Black's first move. At the start of the second move, adjacent Kings are an illegal position.

(However, White may not place his/her King on a square that is adjacent to both Black Kings; because both black Kings cannot be moved away by the end of the 1st black move, which means the 2nd King can attack White's King during the 2nd.)

This, of course, lead to my second discovery: this variant is a bogus game, because this means that White is checkmated upon a check.

If Black's Knight can threaten White's King during Black's first move, it can and will capture him during Black's 2nd move. There must be more than 8 times the regular number of positions for Black to do this than if all of Black's regular pieces are on the board and is only allowed to move once. (Don't quote me on the math; I could be wrong there.)

Unless, of course, we want to enforce a follow-rule that Black may not enter his/her Knight onto a square that would check White's King until it's 2nd move. (It may or may not harmoniously work with the previous rule of White being allowed to be placed adjacten to 1 of black's King and Black having to move that King away during his/her first move. Right now, I'm having logical overload thinking about this. I should let my mind clear, and then come back to build on it later.)

Note that black only has two pieces: a royal King and a non-royal Knight, both double-moving. The double-moving King is of course very much like a (royal) Chu-Shogi Lion, but a double-moving Knight is even far stronger. There is no overlap between its single and double moves, and less overlap between its double moves, so it attacks 40 squares instead of 24.

Note that the rules for moving through check with the double move are not clear.

Black only has one king.

Incorrect; the black king only has to be safe after the second move.

Right, he's in checkmate.

Incorrect again.

I don't see what I was "incorrect again" about. A Knight checking White's King on its first move has the option to move onto the threatened King's square for its second move, thereby capturing him; thereby checkmating White.

Unless there's an extra rule where Black's winning condition is not checkmate but to capture all of White's pieces, which I missed somewhere in the thread?

(While White can checkmate Black?)

That aside, it's what happens when you have information overload: seems that, amidst my grandiose (mis)analysis, I forgot the simple fact that Black could capture a King by being adjacent to White's.

Apart from the brain overload, I guess I was trying a bit too hard to be smart. xD

I also seemed to have misread the OP's post:

 which made me misunderstood the OP stating that there are 2 Kings.

Yes, I made a similar mis-interpretation (thinking that 'king knight' meant a KN compound, i.e. a Centaur). But it was explained later in some of the posts above that he just meant a King-side Knight (i.e. on g1).

And in the first diagram you post here, Bf1-d3 was an illegal move: white is in check, and must resolve that. Interposing is never possible on Knight's checks, so he either has to capture the black Knight (which is not possible here), or move his King out of check (e.g. Ka1).