Well, as you can see, I've barely moved an inch since my last update regarding the Apple Mail problem. It's not clear what is going on yet, but a possibility is that the mail server is having a difficult time differentiating between identical mail clients on different machines (or platforms such as iPhone) Or, perhaps the mail database is corrupted and needs to be cleared out.
Thinking the local desktop computer email account might be corrupted, I tried deleting that, restarting the computer, and then re-creating the mail account from scratch. A group of older messages were downloaded (but not all messages) and only a few of the current day's mail messages were downloaded.
When using Apple Mail on another computer, and requesting to download all messages from the server, only a few of the 178 messages on the server were downloaded. All of the messages should have been retrieved, but they weren't.
When using Thunderbird (a different mail client) to access the mail server and download the messages, all of the 178 messages on the server were downloaded.
It still isn't clear what is going on with the mail server. However, with Thunderbird working, at least messages can be retrieved in some form until the problem is resolved.
Unfortunately, this problem has used up about 4 hours of my morning. For now, I'm going to take a break, go to the gym and do my workout that I didn't get in this morning, and then get a bite of something to eat since I've not eaten since yesterday. I'd planned to check-in at the office briefly this morning, then go to the gym and eat after my workout.