Kf1 in Ruy Lopez

@1

After 7 Nc3, or 7 Bd2, or 7 Nd2 black can take the central pawn 7...Nxe4
After 7 Kf1 Nxe4? 8 d5 wins

The Kf1 move is also sometimes used to avoid exchanging pieces.

When you have a space advantage, it is usually best to AVOID exchanging pieces... because a cramped position is always more congested and more awkward if a full set of chess-men is stuffed into it. Trading pieces tends to relieve the congestion... so the side with the space advantage wants to avoid needless trades.

So White replies to Bb4+ with Kf1 (avoiding exchanges) intending to kick the Black Bishop out with a2-a3 before developing his b1-Knight.

A very similar idea occurs in the French Advance variation.

White anticipates a space advantage, so he avoids any exchanges.

A similar idea also in the King's Gambit: Kf1 to kick Qh4

https://www.chessgames.com/perl/chessgame?gid=1044179 

There are some cases in the Danish gambit where Kf1 is a reasonable move, such as here where you can also play Nc3 or Nd2:

Most of these examples follow a similar theme... the move Kf1 is being played to avoid the exchanges that would follow blocking the check with a Knight.

Ok. That seems way more simple than what my mind was wondering to. One of the main themes is to avoid exchanges. Thank you all so much for the insight! I've got to break these mental barriers that keep rejecting moves like this just because "I want to castle" and other bad foundations that are keeping me from looking at the position objectively and playing what the position is calling for.  I think examples like this will help me break down these barriers.

I'm slowly starting to not dismiss moves just because of how they look. The biggest barrier for me right now is these early king moves in certain positions and trying to discern when these moves are good vs bad. Moves like this seem very important when fighting to keep the advantage.

Try this one:

King's Indian Defense. I'm playing Black against a Titled player (an IM). By playing both h2-h4 and g2-g4 right in the opening, he is obviously trying to just sweep me off the board.

What is Black's best reply? Don't be misled by the "cosmetic" aspects of the move!

I'm happy that I would have played a6 first but I didn't find the d6 square for my knight. I analyzed some continuations with 2.g5 and 2.h5 just to see if a6 was safe and I didn't find any reason not to play it. 

The problem is that I saw Ne8 in some lines but didn't see the plan to go to d6. I didn't pay enough attention to the pawn structure and got caught up in a somewhat defensive mindset. I wanted to attack whites center but was thinking a6 and b5 later on.  

After seeing your continuation, maybe a piece formation could be Nd6, Qe7, Rd8, Bb7, and maybe the rook on a8 is fine for now. I'm looking for a c6 pawn break later on after the b5 pawn break. I would probably try to break on b5 first in this plan since the pawn going to b5 allows Bb7. This all depends on what white does of course. I want to finish my development.

Please let me know what you think of the new plan of mine with your added insight. It helps a lot to see that the knight obviously belongs on d6. I need to work more on not skipping over pawn structure as that is happening way too frequently lately. It is probably due to my current work life though.

It is very similar ( at face value) to Italian game Cracow variation, which i once analysed, but remember next to nothing now.

Engine consideres it to be pretty much an equality, but black is on a quite a minefield, his bishop no longer doing anything usefull and somewhat exposed. I suppose that it is similar in Ruy Lopez.

Only a few things seem fairly certain after computer analysis. The LS bishop belongs on d7, knight on d6, and very surprising to me is that the dark square bishop seems better placed via h3 according to the computer and it is moved there quickly in most of the variations I looked at. I was thinking that after the pawn breaks and everything that the bishop would prove well placed but I guess it is better to take control of the dark squares quickly and that makes sense to me since white doesn't have a DS bishop.

I used SF 15 NNUE and let it get to 30-40 depth on each move.

In this position in the Danish gambit, 4 of 5 games in the Big Database went 7. Kf1. I played 7. Nd2, which 1 of 5 database games went and Stockfish gave ?? and -2.33. Stockfish recommends 7. Bc3.

There are a lot of things happening in that dynamic position.