Tried your set and it played pretty well. Gifs seem to get dropped now and then, but normal gifs are doing the same thing at the same rate for me, so I'm guessing it's not because they are gigantic animated gifs, and just some intermittent bug with loading gifs. I'll see if I can track it down but no promises.
I'm impressed at how well the system is handling those. It probably helps that you only have one per step and there's only 1 animated opponent, and cartoony things compress better than photo-realistic video, but still. And yeah, spam-clicking the portrait is going to trigger multiple loads of it at once, and that's bad juju for just about any machine.
The game does cache images once they load, so that explains most of the behavior you are describing. That's also probably why the images seem to continue animating from where they left off when they cycle back.