It's a hell of a job, but I think I finally got the resolution settings working.
The majority of the work was not really the changing of the screen resolution itself, but the updating of all the things in screen.
When resolution changes I have to update the background size, update GUI positions and scales and also make sure things keep their aspect ratio.

Next to all this I've implemented a working settings configuration file in which the set resolution is saved. All in all a lot of new stuff which is difficult to show you.

At least I can show you a gif of successfully changing between resolutions.

Next I'll keep working on the game settings. Fullscreen/windowed mode is up next.