How much should an average mathematician not working in an area like logic, set theory, or foundations know about the foundations of mathematics?
The thread Why should we believe in the axiom of regularity? suggests that many might not know about the cumulative hierarchy, which is the intended ontology that ZFC describes. (Q1) But is this something everyone should know or does the naive approach of set theory suffice, even for some mathematical research (not in the areas listed above)?
EDIT: Let me add another aspect. As Andreas Blass has explained in his answer to the MO-question linked above, when set theorists talk about "sets", they mean an entity generated in the following transfinite process:
Begin with some non-set entities called atoms ("some" could be "none" if you want a world consisting exclusively of sets), then form all sets of these, then all sets whose elements are atoms or sets of atoms, etc. This "etc." means to build more and more levels of sets, where a set at any level has elements only from earlier levels (and the atoms constitute the lowest level). This iterative construction can be continued transfinitely, through arbitrarily long well-ordered sequences of levels.
Now, for foundational purposes it may be prudent to take the "none atoms" (pure) version of set theory. But in mathematical practice, one is working with many different types of mathematical objects, and many of them are not sets (even though one might often encode them as such). But even if one adds some atoms/non-sets – for example the real numbers –, one does not get a universe which is satisfactory for the practice of mathematics. This is because notions like "functions" or "ordered tuples" are from a conceptual perspective not sets; but we can't take them as our atoms for the cumulative hierarchy – the set of all "functions" or "ordered pairs" ... leads to paradoxes (Russell). Now I wonder:
(Q2) What should mathematicians not working in an area like logic, set theory, or foundations understand by "set"?
Note that there is also the idea of structural set theory (see the nlab), and systems such as ETCS or SEAR (try to) solve these "encoding problems" and remove the problem with "junk theorems" in material set theories. One can argue that these structural set theories match (more than material set theories do) with the mathematical practice. But my personal problem with them is that they have no clear ontology. So this approach doesn't answer Q2, I think.
Or, (Q3) do mathematicians not working in foundational subjects not need to have a particular picture in mind of how their set-theoretic universe should look like? This would be very unsatisfactory for me since this would imply that one had to use axioms that work – but one doesn't understand why they work (or why they are consistent). The only argument I can think of that they are consistent would be "one hasn't found a contradiction yet".