In the world of ZFC + Grothendieck universes, is there a standard model for ZFC?

Question

Consider the set of ZFC axioms extended with the following axiom of universes:

For every set $s$, there exists a Grothendieck universe $U$ that contains $s$, i.e. $s \in U$.

1. Is it possible to prove that, for every set $s$ there is a minimal universe $U_m$ containing $s$, such that $U_m$ is contained in every other universe $U$ that contains $s$?

2. If the answer to the previous question is, in general, "no", can it at least be shown for the particular case that $s$ is the set of natural numbers?

3. If the answer to the previous question is "no", is there a set in the world of ZFC + the axiom of universes that can be considered standard by any other reason, in other words, can we identify, on intrinsic grounds rather than arbitrarily, a particular set $V$ which merits being called "THE class of ZFC sets"?

* If feasible, please try not to involve category theory in your answer.

Answer 1

A Grothendieck universe is the same thing as a set of the form $V_\kappa$ where $\kappa$ is an inaccessible cardinal. So, given any set $s$, there is a smallest Grothendieck universe, namely $V_\kappa$ for the least inaccessible cardinal greater than the rank of $s$.

Alternatively, it is immediate from the definition of a Grothendieck universe that any intersection of Grothendieck universes is a Grothendieck universe, so just take the intersection of all Grothendieck universes containing $s$.

That said, the smallest Grothendieck universe is pretty much never considered to be the "standard model" of ZFC. Why should it be? For instance, if you accept the axiom of universes, why shouldn't you think that the "standard model" ought to have inaccessible cardinals?