When I move the cursor in the editor, the preview does not move. It does not move when I press Ctrl+S either. Even worse, when I save changes in chap-unfixed.tex the automatic recompilation does not happen. I attach the relevant files of the working directory.
Created attachment 117935 [details] the screenshot
It seems from the screenshot that "Live preview" is disabled, which is needed for auto-scroll and compile on save. Enable live preview, either using the button on the bottom of the window below the editor pane (below Next Latex Badbox in your screenshot) or from Build -> Live Preview menu, and try again.
Yes, it helped. Unsure how I reset live preview off however.