The board setup is this:

Black Rook on QN1
Black King on Q6
White King on QN6 and
White Pawn on Q5

[For those who can't read this ancient English notation:
W: Kb6 Pd5
B: Kd6 Rb1]

White has just been placed in check from the Black Rook and it is Whites
move. White moves K to QB6 -- check. This is crazy right? Ahh not so fast.

[i.e. Kc6]

Normally White King could not move within one square of the opponents King
because it would violate the basic principle of placing itself in a position
to be captured. In this case however the White Pawn infers a special
protective relationship to the White King such that it is not eligible to be
captured by the Black King on the next move. The White King, in fact, has
checked the Black King and the Black King must move from check!

Is this a correct analysis of the situation and interpretation of the rules?

Capture of the king ends the game immediately. So it's irrelevant whether the
black king is in check after the capture or not. So your king-move isn't legal.

Joost
--
Du hast mein Herz zerrissen, meine Seele geraubt
Das es so enden würde hätt` ich nie geglaubt [Aus der Ruinen -]
Ohne Rücksicht auf Verluste, hast Du meine Welt zerstört [L'Âme Immortelle]
Eine Welt, die vor kurzem nur uns beiden hat gehört