Can White deliver Fool's mate?

No. The mate needs two careless pawn moves in a row from the same side, and by the time White has played two moves Black has only played one. The fastest mate White can force with help is on move three, for instance 1.e4 f6 2.d4 g5 3.Qh5 mate.

Are there other move orders for Fool's mate?

Yes, a handful. 1.g4 e5 2.f3 Qh4 mate is the same idea with the moves swapped, and 1.f4 e6 2.g4 Qh4 mate works because the f-pawn on f4 leaves f2 just as empty. What every version needs is the f-pawn off f2 and the g-pawn on g4.

Has Fool's mate ever happened in a real game?

Not in any serious game I have ever seen. It requires White to play two moves that no one plays on purpose. It shows up in first lessons, in blitz played one-handed, and in the odd game where someone has stopped paying attention.

Fool's mate: the two-move checkmate

Two moves. 1.f3 e5 2.g4 Qh4 mate, and White has not developed a single piece.

From the starting position: 1.f3 e5 2.g4 and the queen lands on h4 with mate.

Here is the finish. The queen on h4 attacks e1 along the diagonal h4, g3, f2, e1, and every one of those squares is empty because White emptied them.

Checkmate on move two. The white king cannot go to f2, both squares on the diagonal are empty, and no white piece can reach g3 or f2 to block.

What each pawn move gives away

1.f3 opens the door. With the pawn gone from f2, the king on e1 is suddenly sitting at the end of a diagonal that runs all the way to h4, and only g3 stands between them.

2.g4 slams it shut behind him. That is the move people miss. Without it, Black’s …Qh4+ can be answered by g3, blocking the check with a pawn that is defended twice. Push the pawn to g4 and g3 is no longer available to be filled: the pawn has walked past the square it was meant to guard. So when the queen appears on h4 there is nothing to interpose and nowhere for the king to go, because e2, d2, d1 and f1 are all still occupied by White’s own army.

That combination, an open diagonal plus no blocker, is worth understanding on its own. It is the same reason the checkmate you should really fear on move four exists, just aimed at the other side of the board.

Nobody loses like this, and that is fine

I have never seen Fool’s mate in a tournament game and I do not expect to. Two moves is not enough time to lose a game of chess by accident.

So why teach it? Because it is the cleanest demonstration a beginner will ever get of what checkmate actually is: attacked, no escape square, nothing to capture, nothing to block. Set it up, then take the g4 pawn back to g2 and watch the mate evaporate. One square, and the position goes from over to normal.

The habit worth taking away

Be careful with f2 and g2 (or f7 and g7 as Black) in the first ten moves. Not paranoid, careful. The f-pawn move is fine when it has a job, for instance in the King’s Gambit after 1.e4 e5 2.f4, but there the g-pawn is still at home and 2…Qh4+ can simply be met by 3.g3. It is the pair of moves that kills, not either one alone.

If you want to see how quickly this idea scales up from a joke into real chess, work through some two-move mates where the loose diagonal belongs to a properly castled king.