I would probably have chosen to use private messaging from this point so as not to force others to feel that they are intruding upon private grief but, in an altruistic hope of helping others and still with an anticipation of a positive result in the end, I'll stick with the public forum - if you'll all put up with me.
Rather than use the old copy of MozBackup which I've had on file for ages, I downloaded a brand new one. I ran it first to back up my Thunderbird 12 and watched as it chuntered its way through copying my folders etc.
I opened Thunderbird in its own safe mode and I disabled all of the plugins and extensions which safe mode hadn't already done. I then visited global junk settings to switch off all automatic junk filtering. When I visited each account's individual junk setting I noticed that, still, it paid no attention to any changes I made but reverted to whatever it fancied by the time I revisited each account to peek at the consequences of the changes which I had made.
Unable to do more, but at least a little optimistic that the newly reset global rules might hold, I connected up my router and, still in its safe mode, I let Thunderbird loose on any new e-mails which might be waiting on the server while I watched closely. There were six. Two of them disappeared at once while four went fleetingly to my inbox before disappearing from there as well. Junk and deleted remained empty. This suggested that there were at least two separate problems with the operation of my copy of Thunderbird so I decided to go straight to its replacement rather than trying to interfere further because each procedure would require to be tested and that would probably lose me even more new e-mails....
I carefully followed the procedure for uninstalling Thunderbird and then installed it anew, noticing that the new version had moved on to 12.0.1. Then into MozBackup to restore my profile, my face gently wreathed in smiles - well, up until the point that MozBackup posted the message "File missing or corrupt" - that file being the very one which it had itself created only half an hour earlier.
Help, please! This is now very serious.