The "it's not possible" statement in math and the Axiom of Choice

Let me answer in three parts.

1. We when we say that we need to use the axiom is not because for some numbers do know and for some with don't know what is their equivalence class. There are models where the axiom of choice fails and it is impossible to choose representatives from $\Bbb{R/Q}$.

   This is why we need the axiom of choice, and will always need it, in order to construct a non-measurable set such as a Vitali set. It is not about our lack of knowledge about the explicit real numbers and so on, it is simply because we know that it is consistent that it is impossible to do if the axiom of choice fails.

2. Yes. We can prove that in some models it is impossible to make this choice, for example there are models where all the sets are in fact Lebesgue measurable, and there are others where all the sets are Borel. In both these models we have that (1) the axiom of choice fails, and (2) there is no choice function from $\Bbb{R/Q}$.

3. No. It is not the same thing. To see it more clearly observe that the axiom of choice is much stronger than the axiom of countable choice (choice from countable families of non-empty sets). But it is fine to say that you cannot prove the axiom of countable without some form of the axiom of choice. That is to say, we cannot prove it from the rest of the axioms of ZF. It does not mean that we can prove the axiom of choice if we have the axiom of countable choice at our disposal.

   Similarly the fact that we cannot prove the existence of a non-measurable set, or in particular a Vitali set, without some form of the axiom of choice does not imply that we can prove the axiom of choice from the existence of such set.

   The rule of thumb is that if a statement is talking about a concrete set (in our case, $\Bbb R$, or $\cal P(\Bbb R)$ if you prefer) then it will not imply the axiom of choice. Why? Because we can arrange a model where the axiom of choice holds for sets far larger than this one, and then breaks in the most acute way possible.

You might also be interested in: Axiom of choice, non-measurable sets, countable unions

Asaf's answer is of course correct but I think there may be a misconception by the OP that it does not fully address. Not knowing whether $\pi + \mathbb{Q}$ and $e + \mathbb{Q}$ are the same is not the problem, because we could just split into cases: if they are the same, put $\pi-3$ into $S$ and if they are not the same, put both $\pi-3$ and $e-2$ into $S$. This is an ok thing to do in classical logic, and in either case that "step" of the Vitali set construction succeeds.

There are only finitely many mathematical constants that anyone has ever named, so we could just split into cases for each one as above, but that doesn't get us very far. The problem is that every equivalence class $C$ has the form $r + \mathbb{Q}$ for many different $r$ (infinitely many) and when we choose an element of $C$ we are only given $C$—we are not given the term "$r + \mathbb{Q}$" so we can't just make sure to pick $r$ itself based on the notation. The vast majority of equivalence classes $C$ don't contain any named constants like $e$ or $\pi$. So given a "typical" $C$, we know that it is nonempty by definition but have no rule to pick any element at all (which is the situation that the Axiom of Choice is designed to address.)