Assuming the existence of a worldly cardinal in $V$, what is the smallest cardinal in $V$ not in every transitive $\mathcal{M}$ such that $\mathcal{M}\models\mathrm{ZFC}$ and is a substructure of $\langle V,\in\rangle$?
What about the smallest ordinal such that the previous question holds?
Are either of these questions known?
This will possibly be my last set theory question for a while.
It isn't clear to me what you mean when you say that $\mathcal{M}$ is a substructure of $V$, so I will assume that you are just interested in transitive $\mathcal{M}$ in general.
As Jonathan says in the comments, $\omega_1$ is the least cardinal ommited from a transitive $\mathcal{M}$. This is because, if there are any transitive models of ZFC at all, then there are countable ones and $\omega_1$ cannot be an element of a countable transitive model.
The least such ordinal, call it $\alpha$, is countable and is usually described as the height of the least transitive model of ZFC. In other words, it is the least ordinal such that $L_\alpha$ is a model of ZFC. This ordinal is clearly ommited from some transitive model (e.g. $L_\alpha$ itself, by definition), and every smaller ordinal is contained in every transitive model (again, by definition, since $L_\alpha$ is contained in all those models).