Perhaps I'm not being clear enough.
I run Thunderbird on my laptop. When I do a Get Mail in Thunderbird (I have it set to NEVER check automatically) it downloads the message/s to my laptop Inbox and they are no longer on my POP3 server.
If I have not yet downloaded them with Thunderbird, I can view them on my phone or on my N7. If I don't touch the Delete menu entry or the Delete icon as the case may be, I can go back and view them again on my phone or N7. If I DO touch the Delete entry/icon on my phone or N7, the message disappears from the Inbox on THAT device, but it is still on the server. If I refresh the Inbox on THAT device it is empty. If I then to a Get Mail with Thunderbird on my laptop, the message is downloaded and disappears from the server for all devices.
What was happening with the default email client on my N7 is that if I viewed the message and touched the delete icon the message disappeared from the server. If I then did a Get Mail with Thunderbird there was no message to download.
The default email client on my phone (running 2.x) never had that problem, nor does K9 Mail on my N7.
I understand what you guys are saying and I do understand how mail works. Perhaps I should have mentioned that my career was in IT and I know very well how to setup and run test cases and troubleshoot problems.
I only posted this thread to alert other users who might be experiencing this problem and not understanding what was happening. Incidentally, my wife also has an N7 and she was experiencing the same problem.