Both the statement are true, which is fine because we talk about relative consistency here.
If $\sf ZFC$ is consistent then we know how to generate a model of $\sf ZFC^-+\it G\neq V$.
On the other hand, it is consistent that $G$ is generated by power sets iterations from a countable set of "atoms" of the form $x=\{x\}$. Then we can easily generate a permutation model in which this set of atoms is not well-ordered, but every well-founded set is well-ordered.
Indeed we can even require that $|V|=|\sf Ord|$, but then $|G|\neq|V|$ because $G$ cannot be well-ordered.