Is there a metamathematical $V$?

So, my first question is whether or not, metamathematically speaking, we should take $V$ (the proper class of all sets) as a completed whole or not.

This is something that philosophers of mathematics—along with some mathematicians—have thought about. The basic point is, like the old Aristotelian distinction between the potential infinite versus the actual infinite, as one might apply to e.g. the integers, one can distinguish between a potentialist view of $V$ versus an actualist view of $V$. And one can argue for one position as having merits over the other.

Let me point to a couple refernces which go into more detail. For a philosophical treatment, Øystein Linnebo's "The potential hierarchy of sets" is a nice paper on the subject, arguing for a potentialist view on $V$. For a treatment of some of the mathematical issues which arise from this potentialist perspective, "The modal logic of set-theoretic potentialism and the potentialist maximality principles" by Joel David Hamkins and Linnebo is a good overview.

If that is too philosophical, the second is much more mathematical: Does this matter for formalizing mathematics?

The answer here depends on the specifics of your potentialist commitments.

To explain, let me introduce some concepts. A potentialist system is a collection of structures in a fixed language, ordered by a reflexive, transitive relation which extends the substructure relation. If we're interested in doing this for set theory, this means we want structures equipped with a membership relation. This is supposed to formalize a domain which is unfixed and growing; something may not exist in the current universe, but we can potentially expand to a larger one and find the desired object.

There's a natural interpretation of modal logic in this context. Namely, $\varphi$ is possible, $\diamondsuit \varphi$, at a universe if it is true in some extension, and $\varphi$ is necessary, $\square \varphi$, at a universe if it is true in all extensions. You can then ask what modal assertions are valid for a potentialist system, and this tells you something about the structure of the potentialist system. The above-linked paper by Hamkins and Linnebo calculates the modal validities for a variety of set-theoretic potentialist systems, and has further details/definitions.

To now explain the "it depends": Linnebo and Stewart Shapiro have a mirroring theorem which applies in certain cases; see theorem 5.4 of the Linnebo paper linked above. In short, it says that if your modal system satisfies the modal axioms S4.2, then any proofs in the modal realm correspond to proofs in a classical realm, and vice versa. (See the Hamkins–Linnebo paper for a definition of S4.2; the way I like to think about it is that expresses directedness in your potentialist system.) It's a general result, so let me illustrate with an example.

You can consider $\mathbb N$ as a completed infinited whole, but you can also approach it via this potentialist framework. Namely, you can consider the potentialist system whose worlds are $\{0,1,\ldots, k\}$ for all $k \in \mathbb N$, equipped with the appropriate restrictions of the arithmetic operations. Then, a statement $\varphi$ of number theory is true in $\mathbb N$ if and only if the modal statement $\varphi^\diamondsuit$ is true in this potentialist system. Here, $\varphi^\diamondsuit$ is the modal statement you get by replacing every $\forall$ with $\square \forall$ and every $\exists$ with $\diamondsuit \exists$. The point is, in a universe in the potentialist system you may not yet have a witness to an existential statement, but you can find one by extending to a large enough universe.

So if your potentialist take on $V$ includes S4.2, then there's an equivalent system that takes $V$ as a completed whole, so there's in the end no difference for formalizing mathematics. Maybe one system might be easier to work with than another, but you can translate results from one to another.

On the other hand, there are set-theoretic potentialist systems which don't validate S4.2, and as you extend you make permanent choices as to what is true. (For examples, see Hamkins and Hugh Woodin's "The universal finite set" and Hamkins and my "The $\Sigma_1$-definable universal finite sequence". Both of these are variants on Woodin's universal algorithm for models of arithmetic, and if you're interested in understanding them the arithmetic case is a nicer starting point. Hamkins has a nice paper about the arithmetic case.)

So if your potentialist take on $V$ is more in line with these, then it would make a difference for formalizing mathematics, since it expresses something that cannot be captured with a single universe.

To express a bit of a personal opinion: I think an S4.2 set-up is probably more plausible than the situations in the Hamkins–Woodin or Hamkins–Williams papers. In particular, both of those require nonstandard models to work, and many—most?—mathematicians are predisposed to the view that nonstandard models aren't plausible candidates for being the 'true' universe of mathematics.

Kameryn Williams has already given a very good answer, but perhaps it is worth saying explicitly that there is not 100% consensus on the exact meaning of completed infinity (or actual infinity).

Having said that, it is common practice to formalize "$\mathbb N$ is not a completed infinity" with the formal system $\mathsf{ZF} - \mathsf{Inf} + \neg\mathsf{Inf}$ (i.e., $\mathsf{ZF}$ with the axiom of infinity replaced by its negation), which is bi-interpretable with first-order Peano arithmetic $\mathsf{PA}$. With this interpretation, the answer to your question about whether it would make any mathematical difference to deny that $\mathbb{N}$ is a completed infinity is yes. Specifically, we would lose theorems that are provable in $\mathsf{ZF}$ but not provable in $\mathsf{PA}$, such as the Paris–Harrington theorem.

There are alternatives. As Kameryn Williams mentioned, Linnebo and Shapiro have argued for a novel interpretation of "$\mathbb{N}$ is only a potential infinity." Under their interpretation, one does lose some theorems, but not quite the same ones. There was some discussion of this point recently on the Foundations of Mathematics mailing list.

As for what it means for $V$ to be a completed whole, there is perhaps even less consensus about what that would mean. As you suggest, one possibility is to introduce a distinction between sets and classes and to assert that $V$ is the class of all sets. You could then point to the conservativity of $\mathsf{NBG}$ over $\mathsf{ZFC}$ as evidence that such an assertion does not prove any new theorems about sets. However, some might argue that the assertion that $V$ is a completed whole amounts to the assertion that $V$ is the set of all sets, and that of course leads to a contradiction in a well-known way. The conclusion is that $V$ is not a completed whole. This is arguably how Cantor thought about the Absolute Infinite.

(taken from a comment)

To me, the idea of ordinals being a completed infinity contradicts the idea of ordinals (I mean the informal idea of ordinals, that is, that after every "completed collection" of ordinals there should be another ordinal).