After years with no problems, I suddenly can't send e-mails, I get "mail server responded mf-505....", how do I fix?

If you have changed nothing, have not updated, then it may just be a temporary network issue. I suggest doing nothing for a few hours. If you feel a need to do something, shut thunderbird down and reboot. That may fix it. If not, I stiil suggest doing nothing for a while.

Virgin confessed to having e-mail issues in this area at the end of last week, but they say all has been fixed. I wondered if I might fix it by (a) creating a copy profile using MozBackup, (b) deleting Thunderbird, (c) downloading a new Thunderbird, and (e) restoring the profile from the backup. I'll follow your advice for a couple of days, and see if the issue disappears.

Do you trust mozbackup? we have no idea what it does. And it's caused calamities in the past. And are you accomplishing anything, or just creating another problem? Last week, we saw many comcast users frantic about problems, problems that all disappeared within a day. Do what you will. If you must backup, why not consider the thunderbird recommendation at https://support.mozilla.org/en-US/kb/moving-thunderbird-data-to-a-new-computer