Is it allowed to move the King to a square adjacent to the enemy King?
Terrorissimo
Yes, it is. That is a consequence of not being allowed to blow up your own King: the Kings cannot capture at all, so they can safely approach each other. Like in Atomic Chess, where you can even protect your King from direct capture that way.

In FIDE chess, here black to move, Kc5 is illegal, regardless of the fact that the subsequent move from White Kxc5 would also be illegal because the black Rook is guarding c5.
My point is here, that in this variant Terrorissimo, your explanation why Kings can approach each other is not entirely automatically following from the rules already given.
OK, I see. You are basically asking whether exploding the King is pseudo-legal, and only made illegal by the anti-selfdestruct rule. (So that it can still be used to check.)
My original idea was that exploding your own King is not even pseudo-legal (and hence does not deliver check to an opponent King in the blast zone). The philosophy behind it is this:
In the position you give 1... Kc5 is illegal, because 2. Kxc5 does not really expose the white King to capture, despite the presence of the black Rook: it finitses the game instantly, so black will have no opportunity to ever capture anything with that Rook.
This is different when blowing up your own King to destroy the opponent's one. Then the damage is already done by yourself, and not dependent on retalliation of the opponent that he will never get to do because the game ended.
I have an idea for a new Chess variant. Setup and piece moves are as per FIDE rules, except that pieces cannot capture in the normal way. Instead of being warriors that jump their enemies to kill them in combat, the pieces in Terrorissimo are suicide bombers. So they can explode, destroying all pieces in the 3x3 area centered on them (friend or foe, including themselves). This takes a turn, during which nothing else can be done. So you either move one of your pieces to an empty square, or let one of your pieces that stands next to an enemy piece explode. It is not allowed to blow up your own King, even if that would take out the enemy King too.