Hi Bellbram,
this gets curiouser and curiouser.
I'm not sure about Eudora on a Mac...but on a PC, Eudora can be set to use a browser to view mail. This allows HTML mail and images, including animated ones, to show. Maybe your laptop has this viewer enabled but your desktop does not?
Have you tried just viewing those smilies from your browser? You obviously can't edit them from a browser, but any browser should allow you see them move. I'm assuming you can see all the moving gifs here at TS. If your browsers won't display the movements, then I'm totally stumped. That would only seem to leave a possible browser setting that prevents looping. If these little smilies are prevented from looping, then there will only be one instance of movement and then it will stop. The one movement may happen so fast you don't see it and all your left with is a static image.
I'm grasping at straws here...don't give up..
see ya