I know this is labeled as solved, but I'm pretty sure the reason water hurts him so bad isn't because electricity shocks him again but because he's short circuiting. It's simply electricity moving from an area of high density to low density: Cole's body stores a lot of electricity in such a small frame and water can conduct that electricity. So water literally sucks the power right out of him and discharges into the water. That's also why it hurts nearby NPCs. It's the same thing they teach you in school about body cells moving from a crowded area to a less crowded area. (Or at least they taught me something like that in high school.)
And yes, Cole sponge bathes. (This was stated somewhere, can't remember where.) I'm not sure about rain (I can't remember anytime we saw him in the rain anyways, at least not in the games) but showering would do the same thing to him because the distance between water particles is very small so the electricity can be sucked out of him and hop around on all those particles.
