King defense setups

Are there any general considerations to make when choosing between the stairway defense setup with f7, g6, h5 vs f7, g6, h6, Kh7?

Yes.

The key point is that every pawn move weakens some squares and strengthens others. Of particular relevance here is what bishops are on the board and where are they?

The move h5 weakens the dark squares. Conversely the move Kh7 leaves you vulnerable to a light squared bishop pinning the g6 pawn. Here white has a dark squared bishop but no light squared bishop so Kh7 looks better than h5.

Other considerations are the number of pieces the opponent is attacking with and the number of defenders plus possible hooks for pawn attacks.

Here white is attacking with just the queen and bishop while black is defending with bishop, knight, queen and king. So there is no requirement for h5. White is behind in development and it will take several moves before more pieces can be brought into the attack. Meanwhile if black starts with Kh7 that opens up the possibility of a later Ng8 if further reinforcements for the h6 pawn are required.

Furthermore with the queen in front of the white pawns launching a pawn storm is awkward for white. If black moves a pawn to the 5th rank it would make such a pawn storm easier for white by providing a hook.

In general keeping the pawns on h6 and g6 gives more flexibility in the face of a pawn storm. Black can respond to a g5 push with h5 and a h5 push with g5.

One general consideration is that pawns can't move back. If you play Kh7 you can later still switch to an h5 setup. If you play h5 then that pawn will never defend g5 again.

Another thing to consider is that you don't have to play either h5 or Kh7. If you're worried about a mating attack then trade Queens at once with Qg4 as Evergalo suggested. But here (as Brian Towers noted) most of White's pieces are still not developed to attack. So Black can even consider the aggressive counterattack g5. This seems best, winning a pawn at once with tempo, and after the further Qg3, Nxe4; Qf3, f5, it's White who's getting pawn-stormed. No better is Qh3: after the Q-trade White is down a pawn with weak P-structure and should lose the endgame.

No, there is no general rule, neither ...Kh7 nor ...h5 must be favored nor avoided on principle. But there are many specific patterns that may or may not be important in a given position :

Possible arguments for ...h6-h5

• A light-squared bishop for white, especially if it stands on the b1-h7 diagonal, is an argument against ...Kh7, self-pinning the Pg6.
• If White has or can mount pressure against Pf7, your king might be useful in g8 protecting that pawn.
• If you have some activity yourself on the kingside, control of g4 (e.g. to protect a knight jump there) and/or possibility to push further ...h4 and ...h3 can be handy.
• The closer you are from the endgame, the more valuable it is to grab some space on the kingside and keep the king a bit closer to the center.

Possible arguments for ...Kh7

• If White can exploit the g5-square (e.g. with a knight jump)
• If White can exploit the h5-hook to open lines with g4 (e.g. when White has castled queenside)
• If you can counterattack with f7-f5 sometime (then you prefer to have g5 protected and your king as far away as possible)
• If g6-g5 can be a ressource sometime (that happen less often)
• If the Pg6 needs more protection, especially if you want to escape from the scope of a bishop on the a2-g8 diagonal (f7 been pinned means g6 is not protected at all)

In any case, you should make your choice on concrete calculation and specific strategic features of the position.

In your particular position, besides ...h5 and ...Kh7 which both leave you with no particular problem, you could play ...Qg4, putting a stop to any light pressure by opponent, annoying the Be3 after Qxg4 Nxg4 and trying to exploit the small defects in White's pawn structure in an endgame, or , even better, ...g5 winning the Pe4 with a big advantage for Black.