It may not completely fit, but - just to have said that - in Zermelo-Fraenkel Set Theory, there are no classes. Classes are a concept of the meta-theory - a "class" is no object inside any model of ZF, it is a collection of sets that can be described somehow, mostly by a proposition with one free variable $A(x)$.

Its a convenient way to write down even the axioms of ZF, but you cannot quantify over them, and every proof of the form "for all classes ..." I know is just by structural induction on formulas.

ZF itself doesnt say anything about classes directly (except maybe that some are not empty). It specifies a few simple given objects and operations you can do on them to create more complicated objects, and these objects are called sets.

Actually, I dont see the point in such discussions. I mean, the reason for not having general comprehension $\{x|A(x)\}$ is that you run into the well-known contradictions (like $\{y|y\not\in y\}\in\{y|y\not\in y\}\rightarrow\{y|y\not\in y\}\not\in\{y|y\not\in y\}$), so one has to restrict the notion of a "set" so far that he still can do anything he needs, but doesnt produce contradictions. And then classes are kept as a concept for collections of sets that cannot be expressed in this notion, when you need them. I dont see the "philosophical" consequences of this.