and Q2: why do they exist?
good question.
this is a person or people who have been asked or volunteer to help out with coordinating between the teacher and other parents. when a project calls for sending socks and glue, this parent gets to send the emails, at least that's how they do it at the school's my kids attended.
I don't recall a homeroom mom when I was a child, however, so I guess it is somewhat new.
the other big one is: end of year or teacher appreciation gift coordination. I am sure there are other duties as well, but just volunteering in the classroom here and there would probably not quite be the same as 'THE homeroom MOM'.
Why do they exist? I am not sure.