With the third case it accesses the settings file, but does not write anything.

When does it work and when not:

• It works: if I change columns, column widths, sorting column or window size. Then close Kleopatra and restart it within the same environment (screen size).

Kleopatra running on Linux (Ubuntu 20.10, 21.04; Fedora 34, 35 (rawhide)) does this. Closing Kleopatras window saves columns and column widths as shown (it even works if I change the systemwide used font).
On Windows 10 this does not work. Closing Kleopatra via the windows "Close Button" or by selecting "Close Window" or "Exit" from the main menu settings will not be saved. Opening the window again will show columns as they where after installing (way to small for displaying the dates created and expired and the hash of the key). The sorting column is lost too on Windows, but not Linux.
I am unsure if this bug is triggered by my company setup, or if it exists on any Windows 10 installation.