What is that fuzzy gray stuff on one of the rods that makes a "floater" when I remove the cap?

That is pure silver that was captured on the surface of hydrogen bubbles. It can build up enough to make a bubble cluster that gives the appearance of "furry fuzz". Both bubbles and water have a surface tension...an air to gas interface. When you remove the electrodes from the water, some of the silver will transfer from the bubble surface to the water surface as the bubble pops. This can either be filtered out or scooped off with a plastic spoon. The "floater" is extremely thin and contains very little silver. It looks a lot worse than it is.

Removing the cap very slowly in a verticle fashion while the unit is 'on' helps limit the transfer.