Well, if we were somehow obtaining chlorine and hydrogen gases, we would have detected the smell of chlorine in the oxygen. The hydrogen we burned, which was fun.
Well, that ain't really that convincing. Did you actually test for oxygen? The most common way is to get it to relight a splint.

I'm not sure I follow, you're still getting the hydrogen from the water, there's no hydrogen in salt... I'm pretty sure that commercial electrolysis (do they even do that on an industrial level) doesn't use distilled water.
I never said anything about getting hydrogen from salt. All you ever claimed was that salt water would be more 'active'. This could be because of the electrolysis of the salt. With a couple of tablespoons in there, the concentration of salt ions is going to be literally billions times higher than the concentration of hydrogen and oxygen ions in your solution.

I just don't see at the moment how this works, chemically or thermodynamically.
When he burns the water, he is basically breaking up the hydrogen and oxygen, which are both gases. As the hydrogen burns it rejoins with the oxygen to form pure water, and the salt is left behind, I assume still in a solid state. If the energy put in is the same as the energy you get out, which it should be(minus efficiency problems with creating the radio waves) the you basically have free water filtration. Even with some efficiency problems, it may still be better than current methods to remove salt from water.
Minus efficiency problems? You don't realise how huge a factor efficiency is. To turn the heat energy back into electrical energy you'd have to use a steam generator. Steam generators generate almost all of the world's electricity, yet nobody has got them to run at more than 50% efficiency. This is because of the 2nd Law of Thermodynamics; heat is the most natural, diffuse form for energy to be in, and it takes energy to get it back into some more useful form.

It wouldn't work like this anyway. Salt is a classic ionic substance; when you electrolyse the water, the first thing that's going to happen is that the metal component of the salt will build up on one electrode, and the gaseous component bubble of the other.