It is known that countability is not absolute, for Skolem paradox shows that $\mathcal P(\omega)$ can be countable [externally speaking] in some models, while not in others. However, it appears to me that this won't be the case if we restrict models to be supertransitive [transitive sets in which all subsets of all elements in them are in them], since a bijection between two sets is a subset of a Cartesian product between them and models of $\sf ZFC$ are closed under Cartesian products.
I have two questions
Is the countability of a set absolute over supertransitive models of $\sf ZFC$?
If so, what are examples of other properties that are absolute over supertransitive models of set theory but not over transitive models of set theory?