Long shot but try right-clicking the desktop and creating a new icon. See if that icon behaves the same.
When I was younger, one of the juvenile tricks we used to play on friends was to take a screenshot of his Windows desktop. Then delete his icons and make the screenshot his wallpaper. Makes it look like his icons are still there, but you can't interact with them.
Are you able to click on anything? Try click-dragging anywhere on the desktop - a selection box should appear. A common cause of what you're describing is your left mouse button is dying. Try a different mouse if there's one available.