Well guys, it works!
The problem was neither with max nor with the Panda exporter proper, it was with the way I used to run the exporter. There seems to be some confusion regarding how to activate the exporter. I used to run it via a script that came with the exporter, a file named pandaxtools.ms. A pdf file that came with the script also explained how to install it and how to run the exporter. I followed the instructions to the word and it never even occurred to me that there could be another way of running the exporter besides this one. Well, to keep a very, very long story short, despite what is stated in the pdf file, that the script should work with max 8 - it doesn't, at least not in my case and that was to source of all my troubles.
The other way of running the exporter, a way that actually works for a change, is through the standard File->Export option in the main menu. In the dialog that opens, all I had to do was to choose "Panda DirectX (*.X)" as the "Save as type" and the exporter dialog opens. And that's it! Everything works when it opens like that. WME can load the resulting X file perfectly with all the textures and even with all the Biped animations! No export to FBX first or anything like it, the Biped animations are exported as is and play perfectly in WME.
Thanks everybody for all the assistance. I hope this knowledge will also help others avoid what I had to go through...