Can a PSG give a counseling if someone falls out of a run during PT?

Absolutely can, but I'd simply check in with that Soldier and see what's going on. Maybe just a bad morning. If it were a recent trend with multiple occurrences, it might get noted in monthly performance counseling as a concern about that Soldier's fitness level. This is Team Leader / Squad Leader level stuff... If a PSG feels they need to call-out one specific fall-out... sounds like there might be more to that story? Was it a battalion or brigade or division run? Is there standing guidance about formation runs in this outfit?

Anyone can counsel you for anything. It's just a written record of a conversation. A counseling statement doesn't "do" anything. It's not an article 15, it's not even corrective training. It's a piece of paper that says, "this is what is expected of you".

Yes.
Download and read ATP 6-22.1, ADP 6-22, and ADRP 6-22. Start with 6-22-1.
Once you finish those, read AR 600-20.
These will help you understand counseling and Army leadership. You'll have the knowledge you seek.