I'm looking for a way to move the application menu in the window's title bar on the left side as it is illustrated on the next image.
Optionally, I will be happy if there is some way to force such application menu to appear for non-GTK windows as Firefox, Chromium, GIMP, etc.
You can achieve the preferred ordering of window control buttons by running the following:
gsettings set org.gnome.desktop.wm.preferences button-layout appmenu:minimize,maximize,close
The : works as a "separator". You can put any button to the left or right of the : to make it appear on the left or right of the title-bar.
In order to place something in the left corner of the window title bar for all GTK and non GTK windows, you can modify the above command in this way (reference):
gsettings set org.gnome.desktop.wm.preferences button-layout menu:appmenu,minimize,maximize,close
Here is how the final result looks like for GTK and non GTK windows:
