Thanks for the help. Unfortunately:
I have the latest Java runtime (it won't update)
There are no LW preferences to be deleted (I already deleted them, I guess that's why I can't find any Limewire folder under the preferences in the Library)
Graphic settings?
Plenty of RAM, I have 1GB, and 600 free right now.
I repaired the permissions, I even booted from the CD and repaired my system disk.

I had removed Limewire completely. However, upon lauching the 4.12 install, it told me that a newer version was installed. I had trashed Limewire and everything called Lime*.*, and emptied the trash.
I don't remember what I did, but next thing I know is that the 4.12 install wouldn't run anymore at some point (quickly appearing then fading out of the dock). So I installed the 4.13 which worked as always... and failed just like 4.12:

My Limewire Setup Window

My Java Preferences

My Java Plugin Preferences

And.... drum roll..... Azureus works perfectly, as it has always been for 2 months.

Looks like I'll have to turn my mom into a pirate :-)