There are a few problems you seem to be having. First of all, the statement "mathematical logic depends on ZFC" doesn't make sense.
As mathematical logicians, when we study formal systems, we should imagine placing that formal system in a box. The box is full of formulas and deductions in the object language. For example, ZFC is a first-order theory with one binary predicate symbol and a bunch of axioms. It holds a privileged position since we tend to think of it as 'the' formal set theory, but there's no reason we couldn't instead use of MK or NF or other set theories for the same purposes.
Mathematical logic is the act of studying formal systems using mathematical (not necessarily formal) methods, and ZFC is just one particular formal system. The important point is that mathematical logic is not a formal system, and although the statement "ZFC is consistent" is well formed, the statement "mathematical logic is consistent" is not. To claim a theory is inconsistent is to claim that there is a formal proof of false in some formal system. Russel's paradox, for example, can be cast as a formal proof of false in ZFC with unrestricted comprehension.
Without the context of first-order logic, and the collections of variables and symbols that are required to write down formulas and formal proofs, the statement "_ is (in)consistent" is not meangingful. The blank must be filled in with a first-order theory, or more generally, some formal system with a notion of formal proof. You can ask 'are we justified in forming and manipulating these collections?' But that's an informal question. As other users have pointed out, it has very good informal answers, for example, the fact that computers work gives us confidence that we shouldn't worry about doing arithmetic and manipulating strings informally.
In order to answer the question 'is the informal set theory we used to formulate first order logic consistent' either affirmitively or negatively, we must define the notion of consistency, and in doing so use the informal set theory in question. The point, again, is that consistency is only defined in the context of first order logic, where we take these collections as primitive and define consistency from there. In the same way we cannot speak of a simple group ouside the context of groups, we cannot speak about formal consistency outside the context of formal theories.
In short: One cannot provide a formal proof of anything without first defining formal proof! Hence, we must start somewhere and take the collections of symbols in first order logic as primitive, or convince ourselves informally that we are justified in forming such collections.