I wonder if you have been the victim of a practical joke, I say this because what you are describing is similar to a joke a played on a friend recently.
I took a screen shot of his desktop and saved that. I then made a new folder, put all his desktop icons in it and moved that folder to his documents.
I then replaced his original wallpaper with the screenshot I had taken, resulting in the desktop looking the same as before yet completely unable to click any icons as they were simply part of the screenshot image
When I click on the desktop icons as usual, nothing will open.
I have to go to the start menu and click on them to open the programs.