If your layered document has a Photoshop background layer it visibility need to be set off when saving a PNG or Gif file for Photoshop background layers do not support transparency.
Thumbnail may be a bit more complicated than you think. I do not use Lightroom, and only occasionally use the bridge. I believe Lightroom and Bridge keep thumbnails they develop for image files in their library databases and caches. Other programs do not have acces to these. RAW Files and Image files have Jpeg previews stored in then. I believe most application use these previews jpeg to create thumbnails. Jpeg format does not support transparency.
Icon on the desktop are not always thumbnails some are embedded icons in application modules other are icon in dll these may well have transparency. However image file on the desktop have generated thumbnails. However you can create a shortcut for an image png file and create a icon with a transparent background for the png file and change the shortcut's icon for the png file to the transparent ico file you created for it.