If we violate the partition principle and add to $\sf ZF$ the axiom that there exists a set $X$ that has a partition on it that is greater in cardinality than the set of singleton subsets of $X$.

Can $X$ be a standard stage $V_\alpha$ of the cumulative hierarchy?

Can $X$ be a non-standard stage $V_{\alpha'}$ of the cumulative hierarchy?

About the second question what I mean is there is a non-well founded model of $\sf ZF$ such that $\alpha'$ is seen as an ordinal internally but externally it is a transitive set of transitive sets but has subsets of it that are infinite descending membership chains.

weakpartition principle. It is a stronger claim. $\endgroup$well-founded. All sets are. Non-standard models have a different membership relation, which means it is irrelevant what is the underlying set. $\endgroup$always1more comment