I think it's working.
I've re-enabled docker and stopped all individual dockers. After that, I've disabled docker again. Mover started immediately and now I have everything on my Array. So this is the solution - together with correct shares setting.
Any pointers regarding issue that started it all?
I don't think I can trust this disk until I know what caused it.
Oh, and thank you both itimpi & JorgeB for your patience and support.