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.
Less than a part per million. I'm pretty sure the figure you mentioned qualifies as less than a part per million.
Hm... according to some studies, yes it is. Check the studies I linked to, especially the ones from Rey and Pennsylvania State University.
Even if it can only produce more water, that doesn't affect my point. What I'm saying is, the combination of things sometimes has qualities that are different than the qualities of its parts. Not every combination of water is equal, and we can easily see this by looking at different ice crystals.
It is a fallacy of composition to say that: one water molecule is identical to any other, and therefore any combination of water molecules is identical to any other. It's as if I were to say: every individual Tetris square is equal to any other, and therefore, any Tetris board with 20 squares already laid is identical to any other Tetris board with 20 squares already laid. That's just not the case. Similarly, if you say that oxygen is just oxygen and it can never make anything but oxygen... well, oxygen atoms can combine as O2, O3, O4 and even O8, and each one has its own qualities.
So you treat the research which says what you like as trustworthy, and you treat the research which doesn't say what you like as coincidence. Some would call that "counting the hits and ignoring the misses". Some would call that "pseudoscience".
I don't know if homeopathy works or not. You seem to be very sure that it doesn't. I have no idea how anyone can be so sure about anything.
Have a good one.