This is not a German 6th, or any other kind of augmented 6th chord. Those are chords that function as dominant harmony, have an augmented 6th in them, and resolve the aug 6 to a unison on the root of the following chord. The only notes in this chord that could be spelled as an aug 6 are the C# and D#, which could be spelled as Eb and C#. If this was functioning as an augmented 6th chord, then the chord would resolve to a D chord, and both of those notes would resolve to the note D. That isn't what's happening here.
Although the sonority is enharmonically the same as a D#7 or Eb7 chord, it's not functioning that way here. This isn't really functional harmony, it's just voices moving chromatically from I to ii7. The reason for the spelling is that (1) they're trying to spell chromatic motion the way you normally spell it, with flats for downward motion and sharps for upward motion, and (2) they don't want to spell the top two voices as Db and D#, because that would be confusing and hard to read (especially for an accompanist on keyboard), so they spell the right hand so it's obvious what it is.
Is there a chord containing an augmented and diminished fifth?
Yes, in jazz you can have "alt" chords, which are basically a way of talking about tritone substitutions, and the scale you would play over that kind of chord would have both a b5 and a #5. I don't think that's what's going on here. That's because of the style of music, because there's no way that G is heard as the root of this chord, and because a tritone sub leading to Am would have a tritone in it consisting of a G# and a D.