Holy cow - she's looking for a treatise!
I'm gonna treat this in a couple parts, first the aug/dim and then the A6 chords.
Augmented and diminished chords are unique things; they are both completely symmetrical (at least if you're talking about the Âº7 chord, which is more common in usage than the Âº triad). That's important because any note in the chord can be considered the "root"....
C+ = C-E-G#. Each of those notes are two whole steps apart, so those three tones can be C+, E+ (E-G#-Bx) or Ab+ (Ab-C-E)
CÂº7 = C-Eb-Gb-Bbb(A). Each of those is three half steps apart, so the same tones can be CÂº7, EbÂº7 (Eb-Gb-Bbb-Dbb, F#Âº (F#-A-C-Eb), or AÂº (A-C-E-Gb)
The idea that they're symmetrical is important to their usage - because you can go in
to one of these chords as if it's one name, and go out
of it as if it's another. That's called a "pivot" chord.
As you noted, Âº is the natural vii chord in a major key. It's also the top three voices of a V7 chord... G-B-D-F
. So it's common to use a Âº triad or a Âº7 as a substitution for V (if you're using a Âº7, you use a root a half step higher than the original - G#Âº7 has the same three upper voices as G7).
Since you can have a progression like ii-V-I - or Dm -> G7 - > C, we can come out
of G#/AbÂº7 into C major. And since that same Âº7 chord has the upper three voices of three other dominant 7th chords, D7, F7, and B7, we can go in
to it as if it was one of them... so you could have a ii-V-I in the key of E - F#m -> B7 -> E.
Now we've got a handy way to switch keys from E to C smoothly:
F#m -> B7 -> CÂº7 -> G7 -> C
or we could simply go F#m -> GbÂº7 -> C
Augmented chords can be used the same way. A common use of a + chord is as a replacement for the first beat or so of I, as in: G7 -> C+ -> C. You end up with one voice moving from G-G#-G, which is coincidentally the same movement as you get using a Âº7 substitution for G7. Anyway, that can lead you to combining:
Db7 -> Ab+ -> Ab with G7 -> C+ -> C to get...
Db7 -> C+ -> C
Again we've used the symmetrical chord to effect a key change.
You can also use either chord to smooth out the voices. IV+ is a nice example of that - if you're moving V-IV-I, moving from V-IV+-IV-I gives you a chromatically descending voice.
As far as not diminishing the fifth in a major chord or augmenting the fifth in a minor chord, that's true to a point. But it's just because there are usually clearer labels for that set of tones: "agumenting" a Cm would give you C-Eb-G# = Ab-C-Eb... it's just an iversion of a major triad. And "diminishing" a C major would give you C-E-Gb, which is the same as F#-C-E. Compare that to F#Âº7 (F#-A-C-E), and it would usually be considered a simplification of a Âº7.
But whenever minor or augmented triads are part of a larger structure you can have an altered fifth. You could see C9b5 or Cm7+; they just aren't very common.
I'll put this up and start writing about the A6 and N6 chords