The Flying Knight

Other than the two move in one turn rule, everything remains the same. Kings cannot move to adjacent squares because of their 'attack route'. Same applies here, only difference is that the black king has twice the normal radius, so the two kings have to be two squares away from each other at all times.
In this variant, the king is usually only used to limit the other king's movement, it can't give check, just like a normal king can't, either.

I don't get that. Why would the black King have to stay two squares away from the white King? Is that an extra rule? It certainly does not follow from the normal checking rules, as the white King is not allowed to do two moves. So Squares at distance two from it are not attacked (if there are no other pieces). So what prevents the black King from going there?

In fact, if passing through check during the double move is not allowed, I don't even see a reason why the Kings cannot approach each other to the same distance as in normal Chess.

If the black King should not be under attack after the first move of the black turn, this also has consequences when you move two different pieces: when in check, the King cannot go to the square where the Knight stands, as you would have to move the King out of check first.

Okay, no offense to you, but I really don't see how hard it is to understand. This is a pretty simple variant. Go ahead, try it, you'll figure it out along the way.
Everything is the same except the two moves per turn for black.
The king can't go through check, and since the white king would 'be in check' by the black king had it be only a square away (which of course is not allowed), they always have to be two squares from each other.
Thus the black king is only used to minimalise the white king's movement (and of course capturing white's pieces if possible) and can't give check just as a normal king can't in a normal game, they always used to corner the other king, to limit the enemy's movement.
The only possible mates are all done by the knight, and are (mostly) smothered ones.
And of course, if black gets checked, he needs to spend the first move getting out of it and only then use the second as he wishes.

Well, perhaps the problem is that what you say is a bit ambiguous. With 'two squares away' you seem to mean 'one empty square between them', just as in normal Chess, right? At some point you seemed to suggest something different, when you mentioned that the black King's range was larger.

Trying it out will unfortunately not be any help in figuring out how the rules of a variant are. Rules could be anything. Both allowing passing through check and forbidding it will give viable (but different) variants. Obviously when allowed to pass through check, the black advantage would even be bigger.

With reasonable play I don't expect that white would allow himself to be smothered. The most likely progression of a game would be that the black strips white bare with his 'duplex knight' through hit-and-run captures, and then simply mates him in the end-game.

The rules are just these:

- Black gets to play two moves at each and every of his turn.
- Black may use the two move to either move one of the pieces twice, both pieces once, or to 'pass' a turn (which is simply moving a piece back and forth).
- When in check, Black has to spend his first move to react to the threat, and (if it is) eliminated, can use the second move.
- Black always has to use both moves, he can't choose to move only once.

Everything else is the same as in standard chess. The thing with the black king and its 'radius' is that it simply has twice the range as a normal king.
So since the white king would be 'in check' and thus has to respond immediately, this scenario simply doesn't happen, therefore they have to be two squares away from each other, like if the black king pushes the white king away with his 'radius'.

The white king can't move closer to the black king, as it would be 'in check' and moving into it of course is illegal.
Do you see what I meant, now?
Of course this position is clearly illegal and I just made it up in a second, but I think I already made myself clear.

It's pretty important to specify that Black has a kingside knight. If Black has a queenside knight, White has a very limited choice for the first move, because 1... Nc6d4# is mate unless Black plays 1.f3, 1.f4, 1.e3, 1.c3, or 1.Nf6

Also, if White plays 1.f3, there is mate in 2: 1.f3 Nc6d4+ 2. Kf2 Ne6f4#

Having a kingside knight removes the massive advantage of being able to give check on the first move.

The above is pretty clear, but unfortunately you then continue with statements that completely contradict it. There is no rule in standard Chess that specifies the Kings have to stay a cetain distance from each other. Just a rule that says you cannot expose your King to capture at the end of your turn. That they cannot stand next to each other is a consequence from that, not a rule.

The rules you give would certianly allow the black King to be on e3 when the white King is on e1. If you don't want to allow that, you need more than 'just these' rules, because that would be an additional difference with FIDE Chess.

The diagram you give increases my confusion even more. Especially that you state this position is 'clearly illegal'. I don't see anything illegal there. Even with the extra rule that Kings have a 'radius' of 2 squares within which other Kings must not approach, the Kings are at a distance larger than that. So what makes it illegal? What I see is just a position where white is in check, which he will solve by capturing the Knight with his Pawn.

The rule that you have to use both moves offers an interesting new question: what happens if you stalemate yourself on the first move? Like in the diagram below:

Well as you may noticed, English is not my first language, so maybe that's the problem, here. You're right, the correct word is 'consequence', not 'rule', but for some reason I thought that was obvious.
Because of the two square radius of the black king, they can't be closer than two squares, as white would be 'in check' which of course, is not allowed, (just like a normal king can't give check in normal games, I have said the same thing 3 times already).
If black moves his king closer to white king (gives check), white has to respond immediately, and so it gets pushed back over and over again.
I can't rephrase it in any better way.
They have to be two squares away, otherwise white would remain in check.
To the other question, of course, if you move into stalemate, the result is a draw. It's pretty obvious.
It's also worth noting (and maybe this is the root of your confusion) that the black king does not jump. It makes two consecutive moves.

The king-side knight was always called king knight. There's nothing confusing in there.

I think that should be checkmate, because Black is in check at the start of his turn, and cannot safely escape it following the rules of the game (making two moves).

I realise that, just saying that having a queenside knight would make a huge difference to the advantage Black already has.

That's why black has a clear advantage. Not only because of the rules, but because of self stalemating, as well.
I guess if there ever was a tournament with this variant, it'd be banned to do so, but it's legit to force a draw this way.
Of course that diagram is a win for white, because black can't make two legal moves.
Not checkmate, though, but it'd end the game instantly.

It's not a normal king, so it is allowed for the black king to give check (and checkmate) as long as the black king is not in check at the end of Black's turn.

That doesn't mean that they have to stay two squares away. It means that the white king has to move away each time they are one square away, and if it can't, it is checkmate. The black king not jumping has no consequence to this.

No, Black just needs to be out of check by the end of his turn.

I would like to check these rules in reliable source.

If the black king can enter check in his first step or not is crucial for white's strategy and overall game balance.

Sounds interesting, but yes, we'd need to know the full rules as the ones listed leave a number of holes, like the one Polar Bear mentioned.

Please don't throw the knight, they are heavy and someone can get hurt if they get landed on

Black just has to be out of check at the end of his turn.

But black doesn't follow the rules, they live outside of it!

Yes, I see you say so. Now prove it and give us reliable link.

So essentially, in an open board you'd need probably need both rooks and the queen to mate?  The king can cross two bishops' diagonals, etc. and escape mating nets.