The idea is that their electrolysis unit is fed by the generator, which is in turn fed by the hydrogen from the electrolysis unit. To make the contraption somewhat more scientific-looking (and less suspect), they've put two water filters in the loop, presumably to 'purify' the hydrogen.

A reason, though perhaps not theirs, storage. If you have excess wind/solar you could use a hydrogen generator to create enough to provide some minimal power for low production times or overnight.

Yes. Make the water salty to increase the effect.

To save on batteries you can drop the end of a low voltage DC transformer in the water.

Yeah, it's awesome. You can also try different electrode materials (connected to the battery with alligator clips) to make the water turn mysterious colors. Copper is the coolest in my opinion, if you salt the water it will go through multiple oxidation stages over time and change from green to yellow to black. (Oh, and if you use copper and let it react for a really long time, it will form copper tetrachloride and if you mix that with copper sulphate (blue crystals used to kill roots, get it at most any hardware store) you can come up with this crazy stuff that will eat up aluminum foil like some kind of nasty acid but does nothing to human skin.)