Sorry, this won't help you, but I never had this problem with Opera. My speed dial tiles are always in the same order I wanted and never shifted. It must be some preference that not works well.

Go to the adress field and click the red heart, so you can change the logo of the websites, but remember: not all websites will display their native logos, so Opera will display instead a default (written) logo.

Thank you for your diplomatic "Kick in the ass" leocg!
It was an extension, ("Reader View") but in my frustration I forgot to search the obvious places.

Thread can be closed.

I just remarked, that the YouTube website is not responsive. It loads normally, but I can't click on any "clickable" element... It just happens ...nothing!. Videos do not load at all. YouTube isn't blocked by the adblocker and there are no other restraints. Clearing the cache and deleting cookies makes no difference.
Under Safari Youtube works absolutely normal, as desired.
Any hints? (Newest Opera 54.0.2952.71)

You have to define Opera as your Standard-Browser: Open Safari and go to preferences and change the Standard-Browser to Opera. You can load Opera on startup, but it doesn't mean anything, because Safari is the Default-Browser, it will open any site, not Opera.

BTW: Latest Opera version now is 49.0.2725.64.

Not quite sure it's an Opera bug...
Latest version running under High Sierra:

If antialiasing is causing you eye strain why not reduce/deactivate it entirely in Mac OS? With system preferences or a tool like Tinkertool. Here a thread I found:
(see #9) https://forums.macrumors.com/threads/can-anti-aliasing-be-turned-off-in-os-x.258717/

There is some preference in chrome://flags:

Perhaps it's of use for you.

If you were "just playing around" why not delete Opera at all and start at the beginning by installing a fresh Opera and import the bookmarks from Safari?