OK, finally I've worked it out.
So, FIRST, repeatedly push Update button in ESET with Test option. The emphasis is on REPEAT SEVERAL TIMES.
It will show that somehow CM updates to 1112 from 1110. I think it took me 7 or 8 attempts.
SECOND, reboot the computer and do full scan (takes time)
THIRD, reboot computer again and do Restore to prior date to occurrence of the error (in my case 20th September).
Fourth, reboot once again and DO THE SCAN AGAIN, as it will again update itself.
Now, the registry keys look as should and although the version of SS is still same, CM1112 corrected the error.
All icons are manageable as before.
Many thanks to forum members, they helped alot and ESET didn't show practical speed to free us from hassles.
Restore function is safer than to mess with registry and I couldn't force my system to change permissions anyway.