Interesting thread. Good comments!
perhaps that C-5 was meant to be C5 (like a power chord).
 Yes! I overlooked that earlier. I use that notation where needed; "C5" being open-5th power chord; no 3rd. Thanks for the reminder. :-)
I agree with James, "A/A" is totally redundant. It's not wrong in any academic sense, but is unnecessary in any conventional situation. The default is always "no inversion". The "bass note" of chord X major, X minor, X whatever, is always X unless otherwise specified.
"A/B" is certainly a valid chord. :-)
Yep. At a glance, that construct appears to be an A major triad played over the 2nd degree (B) of the chord scale. However, 99,999 times out of 100,000, it fulfills the the usual harmonic function of a B7 chord, because it's not an A chord at all: it's a variation of a B dominant 7th chord. Specifically, it is a B(9)sus4.
(The A major triad up top is the triad based on flat-seven of the bass note, B, which is the REAL root of the chord. In other keys: Bb/C, Eb/F, E/F#, etc. The denominator is the root.)
Rarely will you see B(9)sus4 on a lead sheet. Typically it will be A/B or F#m7/B; both fulfill the same function as B7sus4 (even though none of the three are are literally equivalent).
Both forms are EXTREMELY COMMON V-of-I chords in popular songs. I'd say that way, WAY more than half of all R&B, Soul, and soft rock tunes of the '60s, '70s, and '80s utilize that construct when a five-of-something chord is needed. Sit at your piano, play an Em7 chord with your right hand, a lower A with your left, and resolve it to some sort of D (such as D, D6, Dmaj7, Dm, Dm7, Dm9). Whatever you do, it will probably sound somewhat familiar. :-)
On a guitar, it's the easiest barre chord you'll ever play: Mute the low E, and barre all the other strings at (say) the 5th fret -- and that's it. You've got a C/D (more accurately, Am7/D) which makes a nice five-of some form of G or Gm. The root will be on the A string. (There are other forms, of course, but they are beyond the scope of this comment.)
BTW, the little triangle is shorthand for "major", the context virtually always being a Major 7th chord:
CΔ7 = C major 7th chord, (a C-E-G-B tetrad in the inversion of your choice). :-)
I also agree with James' caution about standards. They exist, and though they are not universal or immutable, it behooves us to adhere to them as best we can. Sometimes, context can reveal the meaning of an otherwise ambiguous operator.
For example The "+" can mean "add" but more often means "augmented" or "sharp" (example: D7+5 is a D7 aug 5th; D7+9 would be synonymous with D7#9).
The "-" is often shorthand for "minor" (e.g. "E-7", an E minor 7th chord) but could also be used to mean:
- "diminish" : e.g. "D-5", meaning "D flat-5th", not "D without a 5th" or
- "flat" ("E7-9", meaning "E dominant 7th chord with a flat9th". Note that in this case, the minus sign means "ADD a flat 9th to the chord".
I hope this is useful, and not too confusing.