Think of it just as you speak with your mouth. You say a sentence then you take a breath. We don’t cut our words short because we’re thinking ahead to take a breath. It’s the same when we play an instrument. When you are working on a piece of music look at the phrases and go as far as to mark (with a pencil) in your most important breathing points to remind you to take a breath there.

No, Ab is the same as G#.
I think you are referring to a key’s relative minor. Yes, every major key has a relative minor, and vice-versa.
So, C major has A minor as it’s relative minor…they are related because neither of them has any sharps or flats.
In your example of Ab (G#), the relative minor is B major.
The theory you can use anytime you need to figure this out is to go up a minor 3rd from the minor key to figure out it’s relative major.
To figure out a major key’s relative minor count down a minor 3rd.
The easy way to know a minor 3rd is count 3 semi-tones from the note you are on. So you are on G# count up 3: A, A#,B
B is the relative major.

Yes. We can call it Ab or G#, same thing. Normally if we’re in a flat key like Eb we would call the 4th chord an Ab, not a G#…we keep the #’s with the #’s and the b’s with the b”s and not mix them up.