Just another comment on your definition of choice. I (and most people probably,) would say a choice inherently implies selection from alternatives, where as your definition seems to simply describe taking an action. I see choice as being inextricably linked with free will, because without free will, there are no alternatives from which to choose.
>Can they really be said to exist?
Yes, we can. We can see the effects of their existance. That a particle behaves in a certain way is proof that there were absolute initial conditions acting on it, even if we can't perfectly determine what those conditions were.
>This statement shouldn't be put forward as fact
Well it's a fact as far as it conforms to our current best understanding of how things work. Should new information be discovered in the future, this statement could be revised, but until then it is as accurate as describing any other natural process to the best of our current ability.
>Either way, it's obvious that statements are not simply true or false in every single case.
Please show a real world example in which something is both true and false. Of course statements themselves can be phrased in ways that they are both true and false (or neither true nor false) but they are logically inconsistent and have no real meaning.
The Law of Contradiction (or Law of Noncontradiction) says that propositions about things cannot be both true and false. It's considered to be one of philosophy's fundamental Laws of Thought. (Incidentally, another one of the Laws of Thought says every proposition is either true or false.)