1) Why do people refer to #5 chords as such, instead of as augmented chords? Don't they have the name notes?
No, a #5 chord technically doesn't exist, so whatever chord you can think of to play here should work, as long as you got the correct bass note. Even though it doesn't technically exist, it does exist theoretically and musically, so I would just use a chord that sounds good.
2) What is the usage of #5 chords? If they are treated like augmented chords, then I see them being used for a transition between I and VIm. For example:
C - F - A F major
C# - F - A F #5
D - F - A D minor
O.K., let me get this straight:
1. You are calling C-F-A, which is an F major chord, a 1 chord, so this puts you in the key of F
2. Now, since you are going to play a #5 chord, the note in the bass should be a C#
3. C#-F-A is an augmented chord, but I'm not quite sure if I would use this as a #5 chord
4. From this point, in order to honestly say that you are playing a legitimate #5 chord, I need to know what note is in the bass
This chord you're trying to use here sounds weird to act as a transition between 1 and 6. What I would do is make that #5 chord a dim7 chord, cuz the natural resolution of that dim7 chord would be a minor chord, which 6 is. I would play this: C#-E-G-Bb
But, if you can better show me how u are using this augmented chord, then that's all good. Those chords sound like a familiar progression, but it uses a 1 in the bass for each chord, like this:
F / C-F-A
F / C#-F-A
F / D-F-A
F / C#-F-A
(repeat)
Even in this instance, it's not a #5 because you have 1 in the bass. It's really just an augmented 1 chord. Then again, if I changed the bass note, I could make it a 3, then it could be 1-3-6 progression, like this:
F / C-F-A
A / C#-F-A
D / D-F-A
(repeat)
I could change the bass notes again, then make it a 1-3-4 progression also, like this:
F / C-F-A
A / C#-F-A
Bb / D-F-A
(repeat)
The possibilities are endless. This is an excellent discussion question, I'm glad you asked it. Let's get some more feedback from other theory experts and see what conclusion we come up with.