Is the inconsistency of $ZFC$ relatively consistent with $ZFC$? Well, maybe this is obviously false (or true), but its sounds confusing to me since I have arguments proving both falsity and validity.

*

*Assume so. Then $ Con(ZFC) \implies \neg Con(ZFC)$ ? So, a false formula. (?)


*BUT, if we assume the consistence of ZFC and put ''$\neg Con(ZFC)$" inside $ZFC$ the resulting theory cannot be proved to be inconsistent, because in $ZFC$ we cannot prove the consistence of $ZFC$ (and then get a contradiction with $\neg Con(ZFC)$). So the claim is true.
I guess on 1) I'm not using the definition of relatively consistence right (?)
 A: ZFC and the inconsistency of ZFC are relatively consistent. By Godel's incompleteness theorem, ZFC proves its consistency iff it is inconsistent.
Suppose we could derive a contradiction from ZFC + (ZFC is inconsistent). Then ZFC would prove itself consistent; hence ZFC would be inconsistent, and we could derive a contradiction from it.
Therefore, if ZFC is consistent, then ZFC + (ZFC is inconsistent) is consistent.
Your argument (1) is incorrect. Just because ZFC + (ZFC is inconsistent) is consistent does not mean that ZFC actually implies (ZFC is inconsistent). And even if ZFC implies ZFC is inconsistent, that does not mean it actually is inconsistent.
A: It's important, in a discussion like this, to distinguish between truth or falsity within a specific model and provability and refutability, which refers to a sentence that's provably (from the axioms) true or false, and thus (thanks to the Completeness Theorem) must be true or false in all models.
Also remember that $\operatorname{CON}(\mathsf {ZFC})$ is a specific existential first-order formula that has a useful interpretation in a standard model; namely, that it quantifies across all possible proofs of first-order formulas.
All we really know is that in a model where $\lnot \operatorname{CON}(\mathsf{ZFC})$ holds, that interpretation fails (assuming $\mathsf{ZFC}$ is in fact consistent); i.e., there is some (non-standard) number that witnesses the failure of the formula we use to represent $\operatorname{CON}(\mathsf{ZFC})$, but that non-standard number cannot be interpreted as a first-order proof.
A: In the language (LOST) of ZFC it is impossible to make sentences about sentences. So we cannot add Con(ZFC) or $\neg$Con(ZFC) to the schema ZFC and still have a collection of axioms in LOST.
We can engage in Godel Numbering. This gives us a structure denoted [ZFC], definable from ZFC. And we can define in ZFC what con[ZFC], i.e.,  "[ZFC] is formally consistent",  means.
What Godel showed is that if A is any recursive & consistent collection of axioms from which you can define and deduce elementary arithmetic of $\Bbb N_0$ (e.g the Peano Postulates), then  con[A]  is not a theorem of A.
Failing to distinguish between A and [A]  results in confusion or nonsense, and is very common among students.
