Is it possible to execute an action by double clicking on the Desktop?

I'm guessing you mean by clicking on the desktop itself, rather than an Icon on the desktop (the latter can be done just by having the script in your desktop folder...)

If you're willing to compromise a little, using CCSM

apt-get install compizconfig-settings-manager

The 'commands' plugin allows you to bind commands (like running your script) to events. Under the 'Button Bindings' tab you can assign your command to when you Click while your mouse is on one of the edges. I appreciate this isn't the complete solution, but as you can select all of the edges, it'll almost do what you want.

Hope that gets you some of the way.

This is completely possible. If you place any program (script or binary) on your Desktop and ensure that it has execute permission, it will be executed by double-click by the vast majority of desktop rendering software (very definitely thunar and nautilus, but anything else that follows the same specifications). Note that some desktop handlers may ask for confirmation that you wish to run, rather than view, the program, depending on the implementation of the program (this is in part to workaround a historical issue that files from FAT tend to all be marked executable, even when one doesn't want to execute them).

If you prefer a pretty icon, create a .desktop file on your desktop referencing the program you wish to run. If this is granted execute permission, any compliant desktop rendering program will launch the program as described in the .desktop file, and should do so without asking for confirmation in any way.

The specification for creation of .desktop files is http://standards.freedesktop.org/desktop-entry-spec/latest/ , or you might start with something already created under /usr/share/applications/ . You can check to make sure your .desktop file is correctly written with the 'desktop-file-validate' program from desktop-file-utils.