King + Pawn vs King

Is it always a win for the extra pawn assuming the lone king can't capture the pawn before the supporting king can protect it? I believe I've had trouble in the past securing the pawn the whole way while trying to avoid stalemate.

Not always a win.. sometimes a draw if the opposing king is between the back row and the pawn

It is a win if the King with the pawn can reach the 7th rank with the pawn in tow safely behind it on the same file, if not it's a draw.

RULES for pawns on files b-g.

White (the side with extra pawn) wins if at least any two of the following conditions are met:

* (a) his king is in front of the pawn
* (b) he has the opposition
* (c) his king is on the sixth rank.

It's a draw if black (the defending side) can get his king to the square in front of the pawn, or the square two spaces in front of the pawn, with one exception. An example of this exception is White King=c6 White Pawn=d6 Black King=d8. With white to move, he wins. This example may be moved horizontally to cover any of the files except when the pawn is on the a or h files.

If the pawn is on the a or h file, it's more difficult to win. If the black king can reach c8 or f8 respectively, then the game is draw because he can either get to the corner, and can't be driven away, or if the white king is already in the corner, keep it there by force. The exception to this rule is when the white king is on b6 (g6), and the pawn is on a6 (h6), and black's king is on c8 (f8). With white to move in that situation, it's a win after h7.

Looks fairly comprehensive to me (at first I thought I had found an exception, but it was not to be...)