You can't make a real transparent background just in Swing... you have to constantly take screenshots using the Robot class. Unfortunately, there is no way for the window to keep up with the background if you are going to drag it across the desktop...
As far as dragging without a titlebar, you'll have to add a mouselistener and convert points on the component to screen points using SwingUtilities.
SkinLF supports non-rectangular windows with transparent areas. (But only on Windows...)