Hmmm, are you sure you didnt set a rock on him accidently, even if just briefly, while moving rock around? Unless you are 100% sure where he is/was the entire time you were moving, it would be pretty easy to put a rock onto the eel and not even know about it.
You could have stirred up something that is staying low in the tank, in the rocks, so now the eel is the one effected because he is sticking his face in it whereas the other fish are swimming around and not getting the full effect. Not likely, but there is a possibility of that anyway.
If you made a major change, like moving rock, and right afterwards the eel started acting "funny" then I'd say something definitely happened there. Either physical damage to internal organs or else some "funk" was stirred up and poisoned him. Like I mentioned earlier, it looks like an animal that has been poisoned.
As far as humane ways to put them down, everyone has an opinion. I personally put my fish into a freezer bag and stick them in the freezer. They just slowly shut down, drop off to sleep and never wake up, that's the way I like to view it. Cold doesnt "hurt" and they are unconscious and then dead long before any body parts actually start to freeze.
Otherwise, you can put it in a bucket and drop in clove oil. A small amount basically stuns the fish (which is useful for when you need to calm a large fish for treatment) and a large amount will kill it.
