Update:
As of October 16th mail is working for me. I have changed nothing other than to have re-entered the SMTP password on one account. All accounts - .ME, other IMAP, GMAIL via IMAP, and local ISP via POP are all working, both sending and receiving.
There were issues of rules having not been executed (email still in inbox that should have been moved to other mailboxes), but that was easily handled. The upgrade to Lion left the rules disabled in mail.app, which I did not notice for a while.
I had been using Thunderbird for 2 months, carefully avoiding the use of Apples Mail.app and all of a sudden it when I tried Mail.app it workred! I am guessing that my several mailservers used that time to bring their software up to snuff with respect to the applicable RFC's. I wish I knew what actually happened, but I don't have the energy to pursue this anymore.
I have since been able to upgrade my iPhone and iPad to IOS 5; my iMac to 10.7.2, and moved from .me to iCloud all around.
Best of luck to everyone else with these problems,
Stuart