Let $\{X_i\}_{i \in I}$ be an indexed family of classes. Do the usual set theoretic operations of intersection, and unions (and all other set theoretic operations) hold for classes as well?
For example is it meaningful to write the following $$\bigcup_{i \in I} X_i$$
and the same goes for intersections? What (operations) can we do with sets that we can't do with classes?
As you probably know, proper classes aren't objects that you can reason about using ZFC. Whenever you write something like $x \in X$ for a proper class $X$, this is another way of writing $\varphi(x)$ for some formula $\varphi$—we can express this by writing $X = \{ x \mid \varphi(x) \}$. So you can only do with proper classes what you can do with formulae.
For example, the universe $V$ is the class $\{ x \mid x = x \}$.
So suppose you had a class $\{ X_i \}_{i \in I}$ of proper classes. Each $X_i$ corresponds with a formula $\varphi_i$. So 'formally' when you write $x \in \bigcup_{i \in I} X_i$, what you really mean is something like $$\bigcup_{i \in I} X_i = \left\{ x \ \middle|\ \exists i \in I,\, \varphi_i(x) \right\}$$ This instantly requires $I$ to be a set, since you can't quantify over a proper class. But in first-order logic it still doesn't make much sense, because you're somehow quantifying over formulae. Another interpretation might be $$\bigcup_{i \in I} X_i = \left\{ x\ \middle|\ \bigvee_{i \in I} \varphi_i(x) \right\}$$ This then requires that $I$ be finite (unless you're working in some kind of infinitary logic).
So in summary, you can make sense of finitary unions and, in the same way, you can make sense of finitary intersections and finitary products of classes.
Regarding other operations: given a class $X = \{ x \mid \varphi(x) \}$, you can make sense of the 'power-class' $\mathcal{P}(X)$ via $$\mathcal{P}(X) = \{ x \mid \forall y \in x,\, \varphi(y) \}$$ Likewise given classes $X$ and $Y$, you can make sense of the 'class of functions $X \to Y$'.
But throughout all of this, remember that the new classes you define typically will not be sets, so you can't reason about them internally.