First we need to assert the general framework. We have a language with relation symbols and function symbols and constants, etc. With this language we can write sentences and formulas.
We say that $T$ is a theory if it is a collection of sentences in a certain language, often we require that $T$ is consistent.
If $T$ is a first-order theory, whatever that means, then we can apply Goedel's completeness theorem and we know that $T$ is consistent if and only if it has a model, that is an interpretation of the language in such way that all the sentences in $T$ are true in a specific interpretation.
The same theorem also tells us that if we have some sentence $\varphi$ in the same language, then $T\cup\{\varphi\}$ is consistent if and only if it has a model. We go further to notice that if we can prove $\varphi$ from $T$ then $\varphi$ is true in every model of $T$.
On the other hand we know that if $T$ is consistent it cannot prove a contradiction. In particular if it proves $\varphi$ it cannot prove $\lnot\varphi$, and if both $T\cup\{\varphi\}$ and $T\cup\{\lnot\varphi\}$ are consistent then neither $\varphi$ nor $\lnot\varphi$ can be proved from $T$.
When we say that CH is unprovable from ZFC we mean that there exists a model of ZFC+CH and there exists a model of ZFC+$\lnot$CH [1]. Similarly AC with ZF, there are models of ZF+AC and models of ZF+$\lnot$AC.
Now we can consider a specific model of $T$. In such model there are things which are true, for example in a given model of ZF the axiom of choice is either true or false, and similarly the continuum hypothesis. Both these assertions are true (or false) in a given model, but cannot be proved from ZF itself.
Some theories, like Peano Axioms treated as the theory of the natural numbers, have a canonical model. It is possible that the Goldbach conjecture is true in the canonical model, and therefore we can regard it as true in some aspects, but the sentence itself is false in a different, non-canonical model. This would cause the Goldbach conjecture to become unprovable from PA, while still being true in the canonical model.
Footnotes:
- Of course this is all relative to the consistency of ZFC, namely we have to assume that ZFC is consistent to begin with, but if it is then both ZFC+CH and ZFC+$\lnot$CH are consistent as well.
The general framework to handle logical systems in which terms can be undefined is called "free logic". However, free logic has been studied mostly in the context of philosophy, and is conspicuous only in its absence in mathematical logic textbooks. This is because the way we handle things in mathematics is just to rephrase the question to avoid the undefined values, and to use vacuous quantification in other settings to work around them.
In free logic there are several semantics, but all of the ones that people usually consider do give a truth value to "$(\forall x \in \mathbb{R})[ 1/x = 1]$". All the usual semantics would say that it is false because it is false for $x = 2$. I spent a while looking into this a little bit ago, and what I learned is that the usual trend in free logic is to assign a truth value to as many quantified sentences as possible. Of course one could define a semantics in which a quantified statement has no truth value if there is a substitution instance that has no truth value (e.g. "1/0 = 1" has no truth value in free logic because $1/0$ is an undefined term; but $(\forall x \in \mathbb{R})[1/x = 1]$ is nonetheless false in all the usual semantics).
My interest in this came from looking at statements such as "$(\forall X \subseteq \mathbb{R})(\forall z \in X) [ z \leq \sup X]$". My opinion is that this statement is erroneous, and has no truth value, because it has no truth value when we take $X = \emptyset$. However, by looking at some references I realized that all of the usual semantics for free logic make this statement true, because they make $(\forall z \in \emptyset)[z \leq \sup \emptyset]$ true as a vacuous quantification. Nevertheless my personal opinion, as yours may be, is that the formula at the beginning of this paragraph cannot be asserted in ordinary mathematics because to do so implies that $\sup\emptyset$ has to be defined. It would be possible, I believe, to write a short paper developing a semantics for free logic that mirrors this, although I have not spent any significant time on it.
Best Answer
One way to make precise the distinction you're trying to make is the notion of a well-formed formula in logic. Roughly speaking this is a formula which is built up from other formulas in a meaningful way, so it can be assigned some kind of meaning and it is meaningful to talk about whether or not it is true. A formula which is not well-formed does not in any meaningful sense have a truth value.
In a suitable formal system for talking about arithmetic operations, the expression $\frac{1}{0}$ is already not well-formed; division $\frac{a}{b}$ should only be well-formed if $b \neq 0$.