Originally, I had written a whole spiel about the Von Neumann universe and Cantor's theorem in relation to the collection of objects that can be described in the language of set theory; but then I realised that the entire issue reduces to picking "some extension of ZFC with the words 'class' and 'size'" and a handful of logical arguments. So rather than try to address every detail, I'm simply going to list the arguments.
Premise 1-1
Every set is a class.
Premise 1-2
Every subset of a class is a set.
Premise 1-3
The powerset $\mathcal{P}(C)$ of a class $C$ contains all and only subsets of $C$.
Premise 1-4
$V$ is the class containing the empty set and all subsets of itself.
Conclusion 1
$\mathcal{P}(V)=V$
Premise 2-1
The size of the powerset $\mathcal{P}(C)$ of a class $C$ is $2^C$.
Premise 2-2
If $A=B$, then the size of $A$ is the same as that of $B$ ($A\cong B$)
Conclusion 2
$2^V\cong V$
Question
What is the smallest class $E$ such that $V<E^V$.
What is the smallest class $E$ such that $\kappa<E\implies \kappa^E\cong E$
Comment: The above is consistent with ZFC provided that the size of every cardinal is less than that of $V$. If every class which is contained in another class is a set, then the above is consistent with NBG.
Comment: The original question touched on $V=L$. Should there be a tag for that, or is it to specific?
There are two notions of "power/function collection" at play here; your post focuses on one, but your particular question implicitly assumes something only true of the other. Untangling these we get two versions of your OP with correspondingly different answers.
Below I'll work in some theory $\mathsf{HCT}$ capable of handling sets, classes, and hyperclasses which is a conservative extension of $\mathsf{NBG}$; whipping up an appropriate theory is easy so I'll ignore the details. Analgously to $\mathsf{NBG}$, all objects in this theory will be hyperclasses, classes will be hyperclasses contained in some hyperclass, and sets will be classes contained in some class.
Let's look at the notion of "powercollection" first for simplicity. There are two ways we could try to define $\mathcal{P}(E)$ for $E$ a class - namely, as counting subsets or subclasses (and it's with respect to the latter that we need to bring hyperclass language into the picture):
When we look at function sets and think about the identification between $2^?$ and $\mathcal{P}(?)$, we run into an analogous "small vs. large" distinction. For a class $E$, the class maps $E\rightarrow 2$ corresponding to elements of $\mathcal{P}_{small}(E)$ are exactly those which are $0$ except on set-many elements. This suggests the following pair of definitions:
We then have:
Per Cantor, there is no surjection from $V$ to $\mathcal{P}_{large}(V)$, and there is a bijection between $2^{V,large}$ and $\mathcal{P}_{large}(V)$ so there is no surjection from $V$ to $2^{V,large}$ either.
On the other hand, we have $\mathcal{P}_{small}(V)=V$. Indeed, it's easy to build injections $V\rightarrow V^{V,small}$ and $V^{V,small}\rightarrow V$, so by Schroeder-Bernstein for classes we get a bijection between the two.
So under the "large" interpretation of your question your Conclusion $1$ is wrong and the answer to Question $1$ is $E=2$, and under the "small" interpretation of your question your Conclusion $1$ is true and the answer to Question $1$ is that there is no such $E$. Similarly, the answer to Question $2$ (restricting to proper class $E$s) is either "No such $E$ exists" or "Every $E$ satisfies the condition," respectively.
(Note that this exactly parallels the "set-level" situation for strong limit cardinals: if $\kappa$ is a strong limit cardinal, then there is a bijection between $V_\kappa$ and the set of subsets of $V_\kappa$ of size $<\kappa$, but $\mathcal{P}(V_\kappa)$ is strictly larger than $V_\kappa$. On that note, keep in mind that strong limit cardinals do provably exist in $\mathsf{ZFC}$.)