is this the fastest checkmate by taking something with a pawn

Yes it is the fastest mate by taking something with a pawn. There is an equally fast variation of this where instead of black playing Nc6 and b6 he plays Nh6 and Nf7 and the pawn itself is the mating piece on gxf7

This could be called the advanced version of fools mate. It just goes to show that moving your king bishop pawn is risky. :]

#2 There are other possibilities:
1 e4 Na6 2 Bc4 Nb8 3 e5 Nc6 4 e6 Nb8 5 exf7#

#3 Moving the king's bishop's pawn is not as bad as people think

Tpr ... As we know...Its not so much about moving the the pawn ...More about being concerned about the weakest squares on the board f2 and f7...guarded only by the King. Moves by these pawns, can be okay in some games...more a matter of when to move them. I can just recall the many times I've ended up regretting moving them. The other players King's Bishop location has a lot to do with it. :]

For the record, without capturing most pawns yield a 4-move solution.

Yes, but the theme here is capturing with a pawn to checkmate.
There are plenty of other possibilities in 5 moves.
1 d4 c5 2 d5 e6 3 dxe6 a6 4 Qd5 Ne7 5 exf7#
1 c4 d5 2 cxd5 e6 3 dxe6 Ne7 4 Qb3 Bd7 5 exf7#

@Abyssinia But there is 1. d4 f5 which is completely sound.

For the record, d4 f5 is not completely sound. As Petrosian said, "If your opponent wants to play the dutch, let him do it". If you really want to play Dutch, you should do it with e6 move order. But then you have to learn French.

Dutch is completely sound. I think you think that 2. Bg5 refutes the dutch, unfortunately this is not true because black can just continue with ...Nf6 instead of messing with the bishop.

Note: I do not like the Dutch myself and do not play it because I do not like the resulting positions. But this does not mean it is not sound.