Theory. Dominant 7th Chords

I walked into class late on Wednesday, and the topic was dominant 7th chords, so figured I needed to backfill a bit.  In particular, what is a dominant 7th chord.

It turns out that the place to start is the major 7th chord.  The major 7th chord is formed by playing the root (1st) and the 3rd, 5th and 7th notes of a major scale (major as in the Ionian mode).

For example,
Cmaj7 = C, E, G, B
Dmaj7 = D, F-sharp, A, C-sharp  (2 sharps in D major)
etc.

To create a dominant 7th chord, the 7th note is just lowered a half step.  Thus,

C7 = C, E, G, B-flat
D7 = D, F-sharp, A, C
E7 = F, A, C, E-flat
etc.

Now that wasn't too hard was it.

But why is it important? Or the same question, in another form; why did Adam then go on to list these, and claim that it'a all your really need to know as a jazz musician.  (Update: I just read a claim that the major, minor7 and dominant chord forms are the only basic forms that exist in Jazz.  Evidently Joe Pass was an ardent supporter of this point of view.)

C7, C7-alt, C7 (sharp9, flat5), C7 (flat 9, sharp 5), and C7(sharp 11, aka, the overtone scales)

When I walked in, Adam was drilling students.

"what's the sharp 9 and flat 5 of C"
"in key of G, what's the flat 9 and sharp 5 of G"

The answers, I think, are..
sharp 9 of C = E-flat
flat 5 of C = G-flat

This is derived from the following
C  D  E  F  G  A  B  C   D
1   2   3  4   5   6   7   8   9

Similarly, the
flat 9 of G = A-flat
sharp 5 of G = E-flat (aka, D-flat)

Key of G contains one sharp:
G  A  B  C  D   E  F-sharp  G   A
1   2   3   4   5   6       7         8   9

Because the context was dominant chords, I assume the final answer is:
C7 (sharp9, flat5) consists of
C, E, G, B-flat, E-flat, G-flat

This looks right, but it doesn't answer either question.

I'll start by confirming the answers at least with Adam next week.