The warnings go out when the site as a whole (mail, files, logs, stats, etc) OR an individual mailbox has crossed the 80% threshold. So in answer to your question, it happens, yes and yes.

You lost me on the mail account overflow part.

"Leave a copy on the server" is something some email programs (read: outlook) made up one day, and is completely foreign to the way POP3 works. I would be surprised if that function works at all.

If you want to leave a copy on the server, setup your mail reader for IMAP instead of POP3 I think you will prefer the results.

As for prevention, I'd suggest a higher quota, use IMAP (which POP3 leave a copy on the server is trying to mimic) and make sure to delete and purge often from the inbox and sent items if you decide to go with IMAP.