Taking a proper class as a model for Set Theory
What is shown in the cases you mention is not that the model is a model of ZFC, made as a single statement, but rather the scheme of statements that the model satisfies every individual axiom of ZFC, as a separate statement for each axiom.
The difference is between asserting "$L$ is a model of ZFC" and the scheme of statements "$L$ satisfies $\phi$" for every axiom $\phi$ of ZFC.
This difference means that from the scheme, you cannot deduce Con(ZFC).
For the proof that Con(ZF) implies Con(ZFC), one assumes Con(ZF), and so there is a set model $M$ of ZF. The $L$ of this model, which is a class in $M$ but a set for us in the meta-theory, is a model of ZFC, since it satisfies every individual axiom of ZFC. So we've got a model of ZFC, and thus Con(ZFC).
Yes, that is true. But note that in its nature statements like $\operatorname{Con}(T)$ are meta-theoretic statements. So when we say that $V$ is a model of $\sf ZF$, we mean that in the meta-theory it is a model of $\sf ZF$.
This is often something which is not stressed enough in introductions to $V$ and relative consistency results: when we prove that $L$ is a model of $\sf ZFC$, we do not "just prove a meta-theoretic result", we in fact prove a stronger statement:
There is a formula $L$ in the language of set theory which defines a class that is provably transitive and contains all the ordinals, and for every axiom $\varphi$ of $\sf ZFC$, $\sf ZF\vdash\varphi^\it L$.
So not only you have this model, but in fact $\sf ZF$ itself prove that each axiom of $\sf ZFC$ holds in $L$.
Let me also share, in my first course on axiomatic set theory, which was given by the late Mati Rubin, we had proved that $\sf ZF-Reg$ and $\sf ZF$ are equiconsistent by practically proving that $\sf PRA$ proves that if there is a contradiction in $\sf ZF$, then there is one in $\sf ZF-Reg$.
Of course, the same can be done with $\sf ZF$ and $\sf ZFC$. And it is much more annoying than using the model theoretic approach. Sometimes with impunity when it comes to class models.