Edit: I see I did not answer all your questions. I am not a logician, so take this with a grain of salt. But basically, in first order logic (in which ZFC is expressed) it is impossible to make circular definitions, and if you can't make one, you can't repair it. The circularity, as I see it, all exists on the meta-level, before you have even gotten around to formalizing the theory. So “breaking the circularity” must in essence happen in the transition between the informal and the formal.
Strictly speaking, first order theories don't even allow definitions at all! What you have to do is to notice that there is a complicated formula NN(x) that we interpret as “x is the set of natural numbers”, and a theorem ∃!x NN(x) in ZFC (where ∃! is short for “there exists a unique …”); then we create a new theory by adding the symbol ω and adding the axiom NN(ω). Now, any formula A(ω) in the new theory can be rewritten in the old theory as ∃x:NN(x)∧A(x), so nothing new has really happened, except for a great amount of simplification.