Thanks, probably Yes, to all of that !
POP, Outlook 2007, "Leave a copy on the Server" etc. etc. Never been a problem, hadn't happened when I read e-mail the day before, and wasn't happening when I shut down, but started immediately when I re-started the next day. Hasn't happened since.
It didn't download e-mails from the server again to do this, we clear those off fairly frequently and Outlook downloaded many e-mails that were no longer on the server, it just seemed to grab every e-mail from every folder and drop them into the Inbox again, but it appeared as if they were downloading, not just moving over from the various folders. Took about 4 hours to shift 1560 e-mails.
Don't use an "Cloud" systems, or similar, don't trust them not to be read by Trump !