I just got back and, so far, the recreation of MessageUidsAlreadyDownloaded3 after deletion appears to have worked. Interestingly, in the first go round, something did go wrong. I deleted the file after closing Mail then reopened. The messages were downloaded as I expected. However, when I closed Mail after that process completed and then opened it again, the mass download happened immediately again. When I went to check the MessageUidsAlreadyDownloaded3 file, I noticed another filed called MessageUidsAlreadyDownloaded3.corrupt with a modified date stamp indicating that it had been the one Mail created the first time after I deleted it. So, something went wrong in the recreation to force MessageUidsAlreadyDownloaded3 to have to be created yet again. I did not notice this .corrupt file in the directory before, so I believe this was the first time (creation time stamp on the .corrupt file was from the time when the first download started, seemingly reinforcing this).
I have opened my other Mail client to let it check for mail on the POP account and did a manual sync on my iPhone. So far, OS X Mail has not re-downloaded. I did delete the dupes but have not emptied the trash in an attempt to minimize variables in case the dupes download again.
For reference, the POP account is a Yahoo! one and the other desktop client is Zimbra. I'd prefer to use Mail only, but Yahoo! will only offer IMAP functionality through Zimbra on the desktop (on the iPhone Yahoo! Mail will do both IMAP with Push). If Zimbra was better, I might use it for all my accounts for the sake of simplicity, but I personally don't like it.
Jeff