Without foundation: Not transitive model of ZFC?

Nice question! The answer is no: Foundation is indeed needed in the metatheory.

Boffa's antifoundation axiom implies that every extensional directed graph $(X; \rightarrow)$ is isomorphic to $(A;\in\upharpoonright A)$ for some transitive set $A$. In particular, any model of ZFC is isomorphic to a transitive model - at least, that's what ZFC-Foundation+Boffa thinks.

Note that the only thing being used here is that ZFC is extensional, so the above also applies to e.g. Quine's theory NF and its variants. Even for non-extensional theories, we get a version of the above result: we get that if $T$ is a consistent $\{\in\}$-theory then $T$ has a model of the form $(A;\in\upharpoonright A)$ for some set $A$.

For a discussion of Boffa's axiom, see Chapter $5$ of Aczel's book.

Show that left cosets partition the group

axiom of choice: cardinality of general disjoint union

Convergence/Divergence of infinite product

If p is an odd prime, prove that $1^2 \times 3^2 \times 5^2 \cdots \times (p-2)^2 \equiv (-1)^{(p+1)/2}\pmod{p}$

$f(z)$ and $\overline{f(\overline{z})}$ simultaneously holomorphic

Find all primes $p$ such that $ p^3-4p+9 $ is a perfect square.

Can $e_n$ always be written as a linear combination of $n$-th powers of linear polynomials?

What is the right way to define a function?

Enestrom-Kakeya Theorem [duplicate]

Updates on Lehmer's Totient Problem

Is there an easy proof for ${\aleph_\omega} ^ {\aleph_1} = {2}^{\aleph_1}\cdot{\aleph_\omega}^{\aleph_0}$?