The chords that we know and love are produced by basically using 1,3 and 5 of a major or minor scale. Thus, Cmaj. is made up from C,E and G. This odd number pattern continues with 7, 9, 11 and 13, the chords named after the appropriate number. So, the Am7 in your example will be A,C,E and G.Since A is the 1, and the root note, and there's a 7 and a minor 3, it gets called Am7. Interestingly, though, by using the same notes, but starting at C, going C,E,G and A, it gets called C6. I really can't understand where 'F/E' comes from. There's no F in it, although playing the bottom string on Am7 will actually make it Am7/E. 'F/E' could be F maj7, as it features an E, but without an F, how can it be an F chord of any sort?
Generally, there are not two different chords playing simultaneously. There could be, for example, C,E.G and B,D,F all together, making Cmaj11, and might(?) be explained as it's C maj. and Bo played at the same time. I can't think of a song, though that would have, say, Cmaj7 and Dm superimposed - that means every note in the key sounding at the same time!
I think you think the slash means to play both the chords on either side of the slash! It doesn't! It means the first chord is the main one to play, and to use the NOTE written AFTER the slash as the lowest note played. That after slash note may, or may not be actually part of the original chord. If it IS, then the chord gets called an inversion, if NOT, then it's just a different note, often used to get from the previous to the next chord. Example - C, C/B, Am7 - which stays with the same basic notes for the chords, but sets up a descending bass line, going from C, through B down to A.
The D6/F# is an example of the former, a first inversion of a D6 chord - being D,F#,A and B, using the F# as the lowest played note.