Truth values in formal systems Given any sentence $A$ of ZFC (or any other formal system, really), we have exactly four possibilities:


*

*$A$ is true and not false


*$A$ is false and not true


*$A$ is true and false at the same time (i.e. ZFC is inconsistent)


*$A$ is neither true nor false (i.e. $A$ is independent of ZFC)

Is my understanding correct?
 A: No. You are making the classic mistake (that put you in good company with many mathematicians) of confusing true and provable.
Truth is relative to a fixed model of $\sf ZFC$. Either $A$ is true in that model or it's not. Even if $A$ is provable, or if it's not provable from $\sf ZFC$. Given a model of $\sf ZFC$ we can verify that either $A$ is true there or it is not.
So the question should in fact distinguish between:


*

*$A$ is provable from $\sf ZFC$.

*$A$ is refutable from $\sf ZFC$ (or disprovable, or $\lnot A$ is provable).

*$A$ is provable and refutable from $\sf ZFC$.

*$A$ is neither provable, nor refutable from $\sf ZFC$.


Terminological issues aside, we sometime confuse the first and second one, by saying that $A$ is true or false in $\sf ZFC$, since if $A$ is provable, it will be true in all models of $\sf ZFC$.
A: If I understand it correctly, you can only speak about "truthness" in some model. As far as I know, the common way how logicians think is that it is either "true in each model" or "true in some model and false in some other model" (this correcponds to your notion of independence) or "false in each model" or "in each model it is both true and false" (inconsistency of ZFC). 
A: What you have done is not but enumerate the "truth table":
$$\begin{array}{c|c}
{\sf true} & {\sf false} \\
\hline
T & T \\
T&F\\
F&T\\
F&F
\end{array}$$
which of course can be done without problem.
However, your terminology is not optimal. I would suggest "tautology" in place of "true" and "contradiction" in place of "false".
