Well, for the mate in 1,
Castling is impossible, surely the white king must have moved to allow the black king there, so there can be no solution. (Also, there must have been a lot more traffic there as well: the black king might have captured the f-bishop on its way, but getting rid of the h-rook takes a much more elaborate manoeuvre.)
For a mate in 2,
Couldn't find a very interesting non-working mate in 2, only many silly ones. Will keep looking.
An assured path to victory:
1. Rd1 - Q-anything 2. Ne4 avoids getting mated by the black queen while forestalling the stalemate that would have been black's next idea. Surely white's material advantage is enough to win after that.
As to the final question:
There's an unmoved pawn guarding every square on the 3rd rank except b3. The pawn on b3 must be the c-pawn, so also b3 has been guarded or occupied by the c-pawn the entire game. There's no way the black king could have gotten past the 3rd rank from its starting square, so the whole position is impossible.
EDITED after the sudden appearance of the corner knight:
In the original problem without the knight in the corner, the immediately preceding moves were uniquely defined:
-1: -- Kh1 0: Ng3+ Kg1 (with some possible captures along the way).
The white knight in the corner removes this possibility, so it cannot be white's turn to move. Since it's black's turn, there is a mate in 2:
1. -- Qxa1+ 2. Bc1 Qxc1#.
In the comment given to all answerers at the time of the edit, OP says he placed the knight at a8, suggesting that the board is the wrong way around. This seems impossible, because then there would be two white pawns giving check at the same time, and that cannot ever happen.
The stuff in the spoiler tags got shuffled around too, so let's recap:
Mate in 1: intended solution: 1. 0-0-0X. Won't work, cannot castle.
Longer victory: intended solution: 1. Rd1 - Q-anything 2. Ne4. Won't work, not white's turn.
Mate in 2: intended solution: 1. -- Qxa1+ 2. Bc1 Qxc1#. (Black wins, sneaky sneaky)
Is it even possible? Well, no. The black king cannot have passed the third rank. Also, "pawns move in the other direction" isn't a plausible explanation.