RE: Homeopathy: Just water?
There's a few incorrect assumptions in your post. If you start with false assumptions, then reaching false conclusions is almost inevitable.
To be clear, homeopathy is rarely 1 part in a million (1:1000000) it's normally around 1:100000000000000000000000000000000000000000000000000000000000 (a single drop in the world's oceans? Not even close to this level of dilution!) so it's no different from water. And one water molecule is identical to any other water molecule; they freeze, vibrate, evaporate, interact etc with well studied consistency. The Fallacy of Composition can't occur when we're combining water+water with nothing else, the only thing we can produce is more water.
Yes, there may still be things we don't know about water, but it curing a disease just because it's been shaken isn't one of them! Numerous high quality research has failed to show any medical benefit from homeopathy. Any perceived benefits are easily attributed to Regression to the Mean, placebo, coincidence etc.