How to formally define the generated $\sigma$-algebra This question is more about the technical part about it's construction than the concept itself. 
Consider $X\neq\emptyset$ and $E\subseteq P(X)$. 
$1)$ Are $E$ and $P(X)$ sets or classes?
In order to construct the sigma algebra generated by $E$ we consider all the sigma algebras containing $E$ and proceed to intersect all of them. The result is the desired sigma algebra.
$2)$ How would we call the family of all this sigma algebras? 
$3)$ When does a family of objects stops being a set and become a class, or whatever is next in hierarchy?
$4)$ Whatever this set of sets of sets is (the family of sigma algebras), is the infinite intersection always defined?
 A: 1) Class is more accurate. $E,P(X)$ are classes of subsets of $X$. $P(X)$ is the classes containing all the subsets of $X$.
2)I do not think there is a name for this family, if you mean the family consists of all sigma-algebra containing $E$.  
3)I am not sure I understand what you said. The intersection never stops in general  as there are infinitely many.  
4) check the def of sigma-algebra and you will find the intersection of sigma-algebras is still a sigma-algebra( even with countable infinite intersection). The family is not empty since $P(X)$ is always in the family. 
A: Even though there is an answer already I figured I'd take a stab one that might look at some of the stuff from a slightly different angle.
First I would like to elaborate on Asaf's comment about classes and sets.
In set theory axiomatized by ZFC, we pretty much always want to deal with sets. Sets are defined in some sense inductively. We say there exists an empty set and then we define operations that can produce more sets. These are: 


*

*Pairing (if you have two sets $X,Y$ there is a set $Z$ s.t. $X\in Z$ and $Y\in Z$

*Union (if you have a set $X$ there is a set $Z$ which contains all the elements of elements of $X$)

*Power set (if $X$ is a set, there is a set $Z$ which contains all the subsets of $X$)

*Replacement (If you have a set $X$ and a formula $\phi$ which acts essentially like a function (i.e. whenever you have $x\in X$ and $y_1,y_2$ such that $\phi(x,y_1)$ and $\phi(x,y_2)$ then $y_1=y_2$. Then all $y$ such that $\exists x (x\in X) \wedge \phi(x,y) $ forms a set.)

*Comprehension (If you have a set $X$ and some formula $\phi$ then $\{x\in X;\phi(x)\}$ is a set).


Note: I fudged in all the axioms but specially comprehension and replacement need to be written in a more complex way. Those two aren't even axioms rather they are axiom schema giving one axiom for each appropriate formula $\phi$.
The main point of these axioms is to ensure that you limit what sets can exist.
Now having said all this it turns out that working just with sets as defined above by ZFC is quite painful. A lot of things we tend to talk about turn out not to be sets. This is usually because we like to talk about ALL of something. ALL groups, ALL vector spaces, ALL rings, ALL ordinals. None of these categories, collections, families or whatever you have end up being sets. The reason is they are too big and if they were sets you would get Russel's paradox right back again. We still like to talk about them in math though so we call them classes or rather proper classes. Classes are essentially a way to talk about entities that could almost be sets but are too big or comprehensive. Actually proper classes are, since all sets are classes, but not all classes are sets.
Given this exposition (and please forgive inaccuracies I was trying to get the point across in at least a sort of reasonable amount of space) we can now answer your first question if we assume that $X$ is a set.
1) Since $X$ is a set then $\mathcal{P}(X)$ is certainly a set. Now all sigma algebras on $X$ are also a set. The reason is sigma algebras will "fit" in  $\mathcal{P}(\mathcal{P}(X))$. One given sigma algebra is also a set since it's just a subset of $\mathcal{P}(X)$.
2) What Brian said. I would just call it the family of all sigma algebras on $X$.
3) This is hopefully answered by the introduction. A family/collection/what have you doesn't really become a proper class. It might BE a proper class if it's too big. This usually means you are trying to talk about ALL of an abstract concept without giving a bounding set. So for example all $\sigma$-algebras on $X$ will be a set, but all $\sigma$-algebras will be a proper class (since you can define a $\sigma$-algebra on arbitrarily large sets).
4) Even if you have a proper class $V$ of sets the intersection of the sets over the whole class will be a set and will be defined (it might obviously be empty though), since the intersection will have to be contained in all of the sets and so in particular in one of them. We can then use the comprehension axiom schema pointed out above.
Edit to respond to question posed in comment:
We need to prove that the collection of all $\sigma$-algebras on a set $X$ is a set as opposed to a proper class.
For this we go back to the definition of a $\sigma$-algebra on a set $X$:
We say $Y\subseteq \mathcal{P}(X)$ is a $\sigma$-algebra on $X$ if $Y$ is closed under countable unions, taking complements and contains $\emptyset$.
Since $Y\subseteq \mathcal{P}(X)$ we have that $Y\in\mathcal{P}(\mathcal{P}(X))$ and we can now apply comprehension to create the set $\mathcal{F}=\{Y\in\mathcal{P}(\mathcal{P}(X));(Y \text{ is a $\sigma$-algebra})\}$. Thus by Comprehension we get that $\mathcal{F}$ which is the collection of all $\sigma$-algebras on $X$ is actually a set.
Notice that for what we want to prove we don't really care about almost anything in the definition of a $\sigma$-algebra on $X$ except the bit that $Y\subseteq\mathcal{P}(X)$ (and since we will be trying to be sort of rigorous we also care that there actually is a definition in terms of a formula $\phi$).
