Is V, the Universe of Sets, a fixed object?
As you noticed, the iterative conception of sets requires a pre-existing universe of sets, and ordinals with which we can label the stages. So if you work within ZFC itself, in other words within an existing model of ZFC, you can perform that iterative construction to obtain $V$. Like Asaf Karagila says here, you cannot get nothing from nothing. Typically, in set theory you work in ZFC, where you have ordinals, and construct $V_k$ for each ordinal $k$. Note that $V$ is not a set, but the entire universe (the one you are working in).
I still think your question is mainly philosophical, your comment to Nik Weaver notwithstanding. After all, you ask:
First of all, the Power Set operation is not absolute, that is it varies between models of ZFC.
Of course, since ZFC proves that $V$ is the whole universe, different models of ZFC will have different $V$. If ZFC is consistent, then it has a countable model $V$. That should not be surprising; one can never 'pin down' the set-theoretic universe, not to say using $V$. The same issue shows up with the natural numbers, as Asaf alluded to in a comment; second-order PA with full semantics does not 'pin them down' because our meta-system MS must always be computable, and hence even if MS has (proves existence of) a full semantics model of second-order PA, there are models of MS whose interpretation of the naturals are not isomorphic (if there are any models at all).
In order to define the Universe of Sets we must begin with a concept of ordinals, but in order to define the ordinals we need to have a concept of the Universe of Sets! So my question is to ask: Is this definition circular?
We cannot define anything, much less whatever is meant by a "universe of sets", without already working under some assumptions. It is not necessary that you work in ZFC, but what other alternative meta-system do you have in mind? Remember that to construct $V$ you need all those set-theoretic operations that you are using, plus the collection of ordinals, and any meta-system that supports all these is going to look very much like ZF or some extension of it.
Boolos noted the same philosophical circularity in this paper (page 15) (which I rephrased to the language used here and emphasized some points):
There is an extension of the stage theory from which the axioms of replacement could have been derived. We could have taken as axioms all instances of a principle which may be put, 'If each set is correlated with at least one stage (no matter how), then for any set $z$ there is a stage $s$ such that for each member $w$ of $z$, $s$ is later than some stage with which $w$ is correlated'.
This bounding or cofinality principle is an attractive further thought about the interrelation of sets and stages, but it does seem to us to be a further thought, and not one that can be said to have been meant in the rough description of the iterative conception. For that there are exactly $ω_1$ stages does not seem to be excluded by anything said in the rough description; it would seem that $V_{ω_1}$ (see below) is a model for any statement that can (fairly) be said to have been implied by the rough description, and not all of the axioms of replacement hold in $V_{ω_1}$. (*) Thus the axioms of replacement do not seem to us to follow from the iterative conception.
(*) Worse yet, $V_{δ_1}$ would also seem to be such a model. ($δ_1$ is the first uncomputable ordinal.)
To put it more cogently, if you take for granted the power-set operation as a primitive, and start with the empty-set, and also take for granted the ability to consolidate into a single operation the union of any iteratively generated sequence of sets, which may itself be used iteratively to generate more sets, then what you can generate appears to be entirely contained within $V_{δ_1}$. And if you additionally allow taking union of arbitrary (countable) and potentially indescribable sequences, then what you can generate is still contained within $V_{ω_1}$.
The crux is that you cannot generate $V_{ω_1}$ without essentially having $ω_1$. And this corresponds to two logical facts: that there is no countable sequence of ordinals before $ω_1$ whose union is $ω_1$, and that $V_{ω_1}$ is a model of ZF with replacement restricted to countable sequences.
More philosophically, if you envision the stages as being generated rather than pre-existing, then necessarily you cannot generate stage $ω_1$ until you have generated all the stages corresponding to countable ordinals. But there is no way to generate all countable stages without having a generation process that already has length at least $ω_1$. And since $ω_1$ does not appear in any stage up to $V_{ω_1}$, you have no choice but to assume the ability to 'run a generation process' of length $ω_1$ if you want to obtain $V_{ω_1}$ and further stages, which implies that the iterative conception cannot give ontological justification for the existence of $ω_1$.
Just to add, it is true that uncountable well-orderings do appear much earlier than $V_{ω_1}$, but the very fact that $ω_1$ does not appear even at stage $ω_1$ (union of all prior stages) should be a warning that one should not consider all well-orderings of the same length to be on equal footing. In particular, to have a well-ordering as a binary relation on a set that makes it totally ordered with no strictly descending sequence is not the same as being able to iterate along it.
Perhaps someone may find a non-circular way to justify ZFC philosophically, but the iterative conception seems to get us no further than countable replacement.
This seems like more of a philosophy of math question than a proper math question. However, in the past Mathoverflow has often been tolerant of such questions.
The basic concern is that the universe of all things there are surely cannot itself be a separate thing. Various responses have been given. On the iterative conception of sets, as it is usually expressed, there is no "completed" $V$, but rather an unending series of stages which are built up iteratively in a process which can never be completed. The obvious objection is that an abstract platonic object is not something which can be "built", nor can it appear in "stages" if it is timeless. One may then be told that the language about building in stages is merely metaphorical, which is not so satisfying.
Russell considered $V$ to be "self-reproductive" in the sense that "we can never collect all of the terms having the said property into a whole, because, whenever we hope we have them all, the collection which we have immediately proceeds to generate a new term also having the said property". Of course the language about collecting into a whole and generation are still dubious, but Dummett reformulated the idea in terms of "indefinitely extensible" concepts where it is one's conception of $V$, not $V$ itself, which keeps changing. However, if you look at it carefully you discover that in order to make sense of the idea of indefinite extensibility you already have to understand the difference between sets and proper classes, which is what the idea was supposed to explain.
I don't think there are settled answers to these questions. Joel Hamkins' work on the "set-theoretic multiverse" is a provocative recent approach which might appeal to you. My view on these matters is given in the last chapter of my book Truth & Assertibility --- in brief, my position is that there is a well-defined class of all sets, but there is no class of all classes because classes cannot be explicitly listed out, as sets (in principle) can. In other words, classes can only be presented indirectly by linguistic expressions, which creates the possibility of there being expressions whose status as representing a class cannot be decided. This means that reasoning about proper classes demands the use of intuitionistic logic and renders the concept "class of all classes" illegitimate. This view is worked out in detail in my book mentioned above.
First of all, I think that part of the confusion stems from talking about "models of ZFC." My recommendation, if you want to sort out what's going on, is to start by forgetting what a model of ZFC is. That way madness lies.
Having cleared our minds of madness, I will agree with you that there is something subtle going on with the very last step $V := \bigcup_{\alpha\in Ord} V_\alpha$. The subtlety derives from the fact that $V$ is not a set. Therefore if you want to conceptualize it as a "fixed, well-defined object like $\pi$" then you should adopt something like NBG or Morse–Kelley, which allows you to talk about proper classes as entities in their own right. The class existence axioms of these systems are what let you define things like the class of all sets without any circularity.
If we insist on using ZFC, then the way to understand the definitions you cited is as follows. You can define ordinals (though not the class of all ordinals) using the set existence axioms of ZFC. That is, you can give a mathematically precise definition of "$\alpha$ is an ordinal" but you cannot define the set of all ordinals (and in ZFC, the only things your axioms tell you exist are sets). Similarly, each individual $V_\alpha$ makes sense. However, the very last step $V := \bigcup_{\alpha\in Ord} V_\alpha$ cannot be conceived of as taking a set-theoretic union to form a new set. You must either think of it as an informal, non-rigorous "definition" or come up with some formal way to handle "proper classes." In Kunen's book on set theory, he adopts the subterfuge of "stepping outside the system" and defining proper classes as formulas. I won't say more because I don't want to go mad; suffice it to say that Kunen's approach lets you stick with ZFC, but at the cost of declaring that $V$ is not a "well-defined object like $\pi$."
There is a "philosophical" way to interpret your question, which is how ZFC can say that there exist sets without first saying that there is a universe of sets and picking things inside of that universe, but I think that this is not really your question.