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$1$ mole of water has $\pu{6.022E23}$ molecules of water, but not $\pu{6.022E23}$ atoms of water, because the expression "atoms of water" has no sense. You are allowed to state that it contains $\pu{18.066E23}$ atoms. But you are not allowed to speak of "atoms of water". Water has not its own atoms of water.


When you compute formal charges you split bonding electrons evenly between bonded atoms. This does not account for differences in electronegativity. It is similar to the difference between formal charge and oxidation state, as well explained in the Wikipedia. A better picture (closer to the real electronic distribution) is, as Pauling would probably suggest, ...


To answer the general part of the question: no, we are not limited to the Cartesian planes and axes. The obvious examples are the ammonia and the methane milecules: the former has several reflection planes, positioned at $120^\circ$, whereas the latter has multiple reflection planes and also the rotation axes (one can easily google both). Regarding the ...


I don't see planes of symmetry through H atoms or OH groups. Actually, these planes are not present. It is easy to check this by looking at the position of the CH3 group after reflection in one of these planes: you find CH3 at a completely different position (none of the original positions is mapped and hence you can distinguish the starting configuration ...

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