$\begingroup$

You asked:

When set-theorists talk about models of ZFC, are they using an informal set theory as their meta-theory?

The short answer is yes. A set theorist is doing mathematics and hence is reasoning informally, just like any other mathematician.

It's important, at least when you're first wrapping your mind around these concepts, to distinguish between formal reasoning and informal reasoning that can be formalized. Formal reasoning consists of formal proofs in some formal system. Informal reasoning includes everything that you encounter in a human-readable mathematical text. Even when you read in a book a claim that "we work in ZFC" or "we work in PA", what you will find in the rest of the text is not a bunch of formal strings, but rather human-readable text written in some natural language like English. What is meant by claims that "the metatheory is PA" is that all the subsequent meta-theoretic reasoning can be formalized in the first-order language of arithmetic and appealing only to the axioms of PA.

Of course, the main reason for informal reasoning is that purely formal reasoning is, except in extremely simple cases, indigestible by human readers. Additionally, in many cases, it isn't particularly important that everything in the rest of the text can be formalized in a specific formal system; what we usually care about is that the reasoning is correct, and that, if necessary, it could be formalized in some well-known formal system. One advantage of leaving the reasoning informal is that the decision as to how exactly to formalize the argument is deferred until that question actually needs to be answered; if you like, it's a form of lazy evaluation.

Occasionally, people squirm when they hear talk about "correct reasoning" without any specification of a formal system. I'm not sure exactly why this discomfort arises. In any other area of mathematics, people don't seem to have any trouble recognizing mathematically correct (or incorrect) reasoning when they see it, even though no formal system is specified. It's only when the subject matter is set theory or logic that some people squirm. Perhaps it's because set theory and logic are perceived to be subjects that are supposed to provide absolute rigor to mathematics and eliminate the need for informal reasoning. It's true that the mathematical community's collective experience has shown that any correct mathematical argument can be formalized in one of a short list of formal systems, but ultimately, we have to rely on the ability of mathematicians to distinguish correct arguments from incorrect ones in order to ascertain which formal systems are appropriate foundations for mathematics and which are not. Furthermore, informal judgment needs to be applied to verify that a particular formalization accurately represents a given informal argument. In this sense, informal reasoning is not entirely eliminable, and one should not expect to be able to entirely dispense with one's ability to recognize correct informal mathematical argumentation.

Is the purpose of ZFC to be used by set-theorists as a framework in which they reason about sets or is the purpose of ZFC to give a object theory so that we can ge an exact definition of what a "set-theoretic universe" is (namely, it is a model of ZFC)?

Different purposes are possible. Set theorists may, for example, study ZFC as an interesting mathematical object in its own right, without caring about its role in the foundations of mathematics.

Of course, historically, ZFC has played an important role in the foundation of mathematics. It furnished an "existence proof": There exists a formal axiomatization of set theory such that all mathematical reasoning can be formalized in it. This "existence proof" then set the stage for later developments such as the "proof" that the consistency of mathematics cannot be proved mathematically—a development that was made possible only because ZFC was specified precisely enough that one could prove theorems about it. And as you suggest, ZFC is a candidate if you want to formalize your meta-theoretic reasoning (though as others have noted, it is typically massive overkill to do so).

So the answer to your question is that yes, ZFC can serve the purposes that you mention, though I would emphasize that those are not the only reasons that set theorists find it interesting.