Hence the qualifier non-A00 openings. a3/a4,h3/h4,f3,g4,b4 are all classified under unusual moves, or A00 under the ECO.
Yes, but you then go on to specifically mention a3, e3, etc. as also being better. :)
Saying f4 is "the worst" of the standard openings is like saying the knight is the worst of the non-pawn pieces, ergo knights are just bad all around.
1.a3 might not be better, but might be. It doesn't weaken the kingside, wastes a tempo (the point however), but also doesn't control the center, but you'll go for that in subsequent moves like 2.Nf3 2.e4 2.d4 depending on how black responds.
Saying a knight is the worst of the non-pawn pieces is actually correct. Even a king is said to be worth around four points in the endgame when you activate it. A knight of course can be stronger than a bishop or even a rook in some specific instances, but overall a bishop is stronger than a knight, not by much, but still is. Heck, the Ruy Lopez Exchange Variation and Bxc3 in the Nimzo-Indian are practically exchange sacs.
The Bird gives some central control, but 1.Nf3! does the same thing without weakening the kingside.
The dirty mating ploys are if white doesn't know how to play correctly and highlight black's attacking potential, activity, development, and access to white's weak squares. Even if white doesn't fall for the tricks black should still be ahead in development and open lines. The From is one really good reply to the Bird recommended by Eric Schiller, but 1...d5 lines also have gambit potential, such as ...h6 then ...g5.
Admittedly, it isn't to everyone's style, some like the King's Indian approach to the Bird, which is fine too. I usually like 1...c5 lines and sometimes like going for the reversed Manhattan, but white doesn't have to allow it, and even if he does both sides still have chances.
Black is a pawn down in the From, true, but in an endgame one pawn usually isn't enough to win unless there are enough pawns (five vs. four or four vs. three usually win, three vs. two and two vs. one can be draws depending on piece and/or king positions). The fewer pawns the greater the drawing chances. However, black will likely have enough piece activity by the endgame to compensate for the lost pawn... if they even make it to an endgame of course.