Note though that this only works because this particular maze has no straight moves available (i.e. intersections where you can't turn but only go forward). For example, using the same rules, if you remove the vertical line directly below the 2x1 block...
Unless I'm mistaken, the result is
Reading through the wall of text, the rules seemed a bit too complicated for it to be "just some random game", so figuring out the magic seemed interesting.
To figure out if the bishop can capture all the lolcats on his first move, we need a "hitbox" for the lolcat; that is, the set of all those squares from which there ...
OK, let's actually take this seriously. As others have said, this is the so-called St Petersburg paradox, and the reason it isn't really much of a paradox is that (1) an extra dollar matters much less when you already have a lot of money and (2) our counterparty may not actually pay up. So let's model that.
The simplest somewhat-plausible way to handle #1 ...
Using only combinations of either single or double button presses:
Using 1 press
Using 2 presses
Using 3 presses
Using 4 presses
Using 5 presses
For anyone curious as to my internal process of coming up with these values, I made a quick chart of each type of press combo and how many variations there were for each type, with x indicating it has ...
The #3 looks like a
The pieces in #1 and #2
Pawns can't be
In #1 the mating move looks like
So the white pieces are
Substituting the known pieces in #2,
Now we just give the black pieces some combination which makes the #2 position a checkmate after that move.
The boards with all pieces: