We are going to remove the arrow of the shortcuts icons first. Hit Windows + R and type regedit. Navigate to HKEY_CLASSES_ROOT\LNKFILE and delete the entry IsShortcut in the left pane. Next do the same at HKEY_CLASSES_ROOT\PIFFILE, et voila the arrow is gone.

Making the desktop icons transparent is far easier. We are going to use the freeware Iconoid to do this. Start Iconoid after installation and click on disable drop shadows in the color tab if that button is available. The default setting is that the icon background is transparent which is exactly what we want. You might need to change the default text color as well if you are using backgrounds that have a similar color than the default font color.