Castling Vertically? Real Rule? If so, how did/does it work?

If it was a pawn on the 7th rank that was just promoted to a rook and then black made some other move that did not go in-between the King castle through move. then you can castle vertically. If it was a rook promoted on another square and then moved there on e8 then you cannot castle vertically. 

Exactly, as rooks have to be on their starting squares strictly.

The big question is whether a promoted rook is considered to have been moved or not !?

no. the e7 pawn can promote to Q, R, N, B like the usual rule. But if you are aware that your king has not moved from its original starting spot and you promote your pawn on the 7th rank to a Rook and your opponent does not have the ability to block the Vertical Castle where after your promotion black gets his move. Then you can Castle Vertically by moving the King first to Ke3 and then Moving the Rook to Re2. It was allowed in another match. It is very rare, but possible. The drawback because it is so rare is the Tournament Director not knowing the rule would rule against it. But the Tournament Director has at his discretion the ability to accept that rule or deny it. The situation is interesting because every player should immerse themselves in rules and have a better grasp of the rules than your opponent and that as any increased experience should give you a legitimate advantage. It is like asking your opponent for a draw. If they hesitate. They probably don't know the position is a draw, but you do. If you ask for a draw and they answer you pretty quickly that means most likely they know the position very clearly for a win. Again, and there is advantage in knowing this in mastering your own standard in the game of chess.

I don't know if it is really considered castling because the whole point is to protect the king, and castling towards the center does nothing but expose the king..

Dont the rules of chess say that castling has to be on the first rank? I think that would rule out any other option. 

Unsure, it just says as long as the king and rook have not moved, and castling doesn't take your king out of, through or into check. I have always been confused on why you can't castle "through" check:

Here the king is moving through the dark squared bishop's diagonal, but the move is legal. (KD3-C4)

wikipedia has the rules of chess and for castling one of the requirements is that it must be done either queenside or kingside. I would think that rules out any possibility of doing it vertically (or diagonally) Also, I think there is a specific rule that says it must be on the first rank, I dont know the number of the rule though. 

btw, the king isn't moving through the bishops diagonal. If it were, then that would mean the dark square bishop has control over light colored squares. But it doesn't. Maybe the point where the two light colored squares meet (and dark squares meet) is infinitely small, and the king passes through some dimensional area that is neither dark or light. 

But it's probably more like a 4 way stop at street intersection. The king can pass through the intersection on the light squares just as easily as the dark square bishop can. They just cant do it at the same time. Right of way is given to the person whose turn it is. 

What you may say about can be legitimate. But in one of Tim Krabbe's chess matches this actually happened but with a different board position but same situation. The TD ruled it was legitimate. Because it is very rare, people will tend to not accept. Why because we play chess and rules, players,  are pretty concrete and set in how the game is played. But this development is not really earth shattering. En Passant and how the Queen evolved in its maneuvers really shaped chess to what it is today. And those situations are more common than a very rare vertical castle. Infinite possibilities in the game, but you must always be aware; or just be like all the rest. Knowing rules, knowing your positional experiences, helps your game against whomever you play.

The thing is that vertical castle is so rare that it doesn't even matter if was legal or not. It happen so rarely in practise. Probably on par bishop promotion if not close. In short: Once in a blue moon.

"But in one of Tim Krabbe's chess matches this actually happened but with a different board position but same situation."

This was not a chess match. It was a constructed puzzle he created to abuse a poorly worded rule (that would likely never would have been allowed in a tournament, and the wording was updated shortly after).

Even so, the king is not actually landing on that square, so why does it matter that it is being controlled? Now that I think of it, not being able to castle out of check makes even less sense. With your reasoning actually, I could say that the king in this position would be only moving along the edge between d1 and d2, and both corners on either side, never actually passing through a square being controlled by the queen:

I think it matters because the king isn't like a knight, which can jump over controlled squares. The only way the king can get to c1 is by way of d1, which is controlled by the black queen. The queen also controls the edge between d1 and d2. So there is no way for the king to get to c1 without passing through a non safe square. I look at it like a piece controls ALL of the square, including the edges.

The intersection between 4 squares is the twilight zone, nobody controls them and pieces like queens, kings, bishops, and pawns are allowed to pass through them freely. I wonder if there are any official rules on those intersections. Probably nobody has thought to challenge it since that's not a place that can be occupied, only passed through. 

And that black bishop controls the corner between c4 and d3 in my last diagram.

In my opinion no. Nobody controls the corners. Chess and checkers would be very different games if pieces were allowed to control corners. Imagine how complicated a game would be even after the opening 4 moves if pieces controlled corners. 

To say the bishop controls the corner between c4 and d3 is to also say the opposing king controls it. It's the same corner. 

There is a variant of chess I believe where pieces can move along edges.

You can't castle with a promoted rook because it technically has moved, but as a pawn.

GM Igor Smirnov's video on vertical castling