This is a partial answer, it only answers the first question.
Working in NFU + Infinity + Choice + $|U|>|P(V)|$ [which is consistent relative to ZFC]
Since we have choice then there is a function $H$ from a partition $K$ on $U$ that has all of its pieces (i.e. elements of $K$) equinumerous to $|V|$, to $P1(P1(V))$, i.e. to the set of all double singletons.
Now we form a surjective function $J$ from $V$ to $P(V)$ such that every set is sent by $J$ to itself, while every Ur-element (an empty object other than the empty set) is sent by $J$ to a singleton set in such a manner that each Ur-element $x$ is sent to a singleton $y$ by $J$ (i.e. $J(x)=y$) if and only if there exists a piece $p$ of $K$ such that $x \in p \land y \in H(p)$.
Now we define a new membership relation $E$ as follows:
$y \ E \ x \iff y \in J(x)$
Now the structure $\langle V, E \rangle$ would satisfy all axioms of SF. But Extensionality would be violated over all and only singleton sets in such a manner that each singleton set has $|V|$ many copies.
This way every set in that structure would be subnumerous to the set of all singletons of its elements.
IF instead of $H$ sending $K$ to $P1(P1(V))$ we let it send $K$ to $P1(V)$, and run the same argument, the result is that we'll get a stratified theory in which every object in this theory has $|V|$ many co-extensional copies!