Getting ready to take a lunch break and go to the gym.
The problem at this point seems to be isolated to the Apple Mail program not properly accessing the mail server. Or, perhaps, the mail server not being completely compatible with the Apple Mail program. The problem just surfaced in the past 24 hours which coincides with the configuration of the replacement iPhone (which involves configuring email on the iPhone).
At this point, new messages from most people seem to appear in the Apple Mail program, but not all messages. Messages originating from the iPhone (such as live feed messages) show up on the server, they show up on the iPhone, they show up in Thunderbird, but these and other messages are not retrieved by the Apple Mail Client on the desktop machine. The problem persists even after deleting all messages from the server and starting all over again with new messages. So, apparently something is corrupted with the account on the Apple Mail client and the corruption remains (and returns) even after deleting the account and creating it again. This is to be expected since many settings and hidden files typically remain after removing an account or program from a computer.
The solution will probably be found in calling Apple technical support and manually removing all remnants of the deleted account, then creating the account again in the Apple Mail client.