Things are progressing here - albeit slowly. Of course my email programme was on the dead drive (one of the very, very few programmes not on the newer C: partition on a different physical drive), so that had to be re-installed and the data backup restored. This happened successfully (yay!) so I can now access my email archive again, and, even more importantly, work on the mails properly and not just in emergency mode over the mobile phone.
There's a solid chance that the new HDD will arrive today, and then I can power down the machine here, put the new one in, and do a proper restore of everything that lived on the old place. Well, apart from the D: drive, which was not included in any backup (causing the issues with the mail programme, for instance) and I sincerely hope that there was nothing important stored on that drive, because it's dead as a doorknob.
On the upside - I have found out what caused the excruciating slowness and proneness to errors with my backup.It was an overenthusiastic Antivirus programme, which decided that since the RAID that the backups are stored on are not an internal drive, everything on them might be dangerous and contain a virus and thus has to be checked very, very thoroughly. Which, obviously, will take time...
So from now on I will try remember that, and if something concernes an external drive and then takes ages, I'll look for the antivirus as a possible culprit. If you're running into a similar problem on your computer, you can either turn off the antivir or, if using the Windows Defender antivirus (which, according to reports, is about as good in getting the bad guys as the free versions of standalones are) there's the possibility to exclude folders from getting scanned.
Oh, and because it bears repeating: if you haven't made your backup yet... well, you know. There's a lot of free and even more affordable programmes for backups out there; some external HDDs come with backup software pre-installed, so you should be able to find a solution without too much trouble. In case you don't want to look about that much at all and already have a drive of five lying around that you can store stuff on, I've used SyncBack Free for a long time and can recommend it.
I will be off travelling north tomorrow, and will return on Monday, so there will be blog silence until Tuesday... when I will hopefully also be able to work from a restored drive.