I don't know how to find the default window manager application name (executable file name and path). I need help to find the window manager.
The default window manager is called compiz. Open a terminal and type
to get the full path. Type
to see in which packages it is replaced. Unlike what was suggested in a deleted comment, metacity is not responsible for Compiz window decoration, but a Compiz plugin called aptly "Decorations".
If, however, you log in using the "Unity 2D" option, you will be using a different window manager, called metacity.