1. ..Nf6 is the Alekhine Defense
not the hottest opening in the book but definitely respectable. It is occasionally played by players like Ivanchuck (world top 10), which says enough. The usual idea is to tempt white to set up a bigger center than he can maintain.
2. ..Ne4 is indeed offbeat, but it still has a name "mokele mbembe".
I can't argue - a very rare move, but..
the game explorer features 8 players rated over 2200 (one even 2500+) playing it and it doesn't score that bad (though certainly not great): 4 wins for white, 3 draws and one black win.
3. ..Ng5 is indeed ??. Black just gives up the knight, total blunder.
4. ..f6 is also very unwise, but its not as bad as ..Ng5, dropping a knight in one. A big ? or two small ??'s.
5. ..gxf6 must be ?? again, mate in one.
All in all you played it well, for the 6 moves the game lasted. 
The first move is definitely not 'the worst' though, and the 2nd move one could debate that it might be a good 'out of book' move to surprise an unprepared opponent. (even though it may objectively be weak).
Just finished a game where my opponent made every worst move possible. Took 6 moves to finish.