Well-foundedness is part of what I mean by "set". (Extensionality is another part.) And ZFC summarizes some aspects of my intuitive conception of sets.
Certainly other sorts of things exist. For example, I exist, and I'm not a set. You ask when it's natural to hold that entities of certain sorts exist. I'd find that natural to the extent that I have a clear intuitive concept of those entities. And I find it natural to work with theories that describe such entities. For example, I have no clear intuitive picture of any entities for which I can see that the axioms of NF are true, so I don't find it natural to work in NF (even though the axioms of NF are nice and clean).
It is a rather remarkable empirical fact that, when I have a clear intuitive concept of some mathematical entities, then I can usually code those entities as sets, perhaps not perfectly but well enough for most mathematical purposes. The same seems to be true for other people's intuition as well, and it seems to be why most of us accept ZFC (perhaps plus some large cardinal axioms) as a reasonable foundation for mathematics.
I don't know whether this empirical fact is a psychological one (about my "clear intuitive concepts") or a historical one (about mathematical concepts that people happen to have introduced) or something else.
By the way, "perhaps not perfectly" above was intended to cover my doubts whether, for example, the standard set-theoretic view of the real line as a collection of points really matches the intuition of continuity underlying $\mathbb R$. It matches well enough, in the sense that we can do analysis using the set-theory version of $\mathbb R$, but I'm not sure that it really matches the full intuition.