When the static discharges happen, you and your cat are acting as transmitting antennae, and radiating low-energy broadband radio noise. Something in the maze of cables is acting as a receiving antenna and picking this interference up.
Sounds like time to upgrade to some cables with better shielding, and/or add some ferrite rings to the cabling where they enter/exit the various components!
You could try selectively disconnecting the desktop and laptop connections to see if one or the other is the culprit.
Interesting experiment: Try grounding yourself to the desktop case while petting the cat. Does that make it better, worse, or have no effect?
Edit: Geeky trivia: You can get an NE-2 style neon lamp
to flash with static electricity. I remember playing around with this as a budding electronics/computer geek when I was young.