Hi,
I will deliver just a quick roundup of what has happened after we fixed the problem - maybe this will help others who experience similar issues.
So the system was working again but some Windows 10 Apps did not work anymore (for example: Photos and the Windows Store Service - yes, the store itself was okay, but I could not update apps anymore, the corresponding service started and exited right away) and re-registering the AppxPackages did not help at all.
SFC and DISM showed no errors at all, chkdsk finished without noticing anything. Every diagcab I used so far in this thread finished without finding any errors.
Besides that the system event log was flooded with DCOM-errors, a few Apps-related errors, services not working, etc.
To sum it up: I could use the system for everything I needed day to day, but something was still wrong.
So I made a backup of my PC and used the Media Creation tool to download the Windows 10 ISO and did a repair installation.
After that the photos app still didn't work, but registering the AppxPackages again worked this time.
Did another backup of the PC and flashed a new firmware on my SSD and in GSmartControl the pink background-color of the soft read errors disappeared from the SMART-parameters, too. I did an extended self-test and this time the test took as long as the program's estimated time of this test instead of a few minutes the last time.
I guess everything is back to normal again, but still I monitor my system closely.