I'd like to add a few words on a different level than the ones very well discussed in the question and the answers so far, although perhaps you've already automatically discarded it: under my dual conceptualist/formalist point of view, I'd say that conceptually speaking, okay, the question is complex, and analyzing it may go in various directions as we can see in the other answers. But well, formally speaking, and I'd take metamathematics to be formal first, there are no sets, only terms for them, and no classes, let alone proper ones, only mere (potentially recursive) enumerations of syntactic gadgets or combinations of those... And as there are way too few syntactic witnesses for every conceivable set in the multiverse, there can be no completed V in that sense; that, I guess, is the only one that 'matters for formalizing mathematics', as in your second interrogation (we could fall back on some unary predicate defining the symbol V in some language of sets or classes, but that's not really absolute, since it's dependent on the ambient theory in this language that we definitionally extend this way).

The story for N along those lines differs, since there's a formal numeral for any conceptual natural number, whenever a numeral system is fixed (all being isomorphic); and the (meta)mathematically speaking best one might be the Peano numerals, namely sequences of the symbol $s$ ending with an appended $0$ (as finally, arithmetic boils down to structural properties of such strings). And those might be collected in a grammar recursively enumerating them and only them, say in Backus-Naur Form, as follows:

$$n::=0\mid sn$$

That's the only actual completed whole of actualized natural numbers I'd see, and I think there's no real doubt of its usefulness to formalize mathematics (if only because of recursive definitions of functions, inductive proofs of universal propositions...).