(Answer for 2.) One reason most mathematicians can avoid the theory of proper classes is through the use of Grothendieck universes. In brief, one chooses a strongly inaccessible cardinal $\kappa$, which is the cardinality of a set so big that it cannot be constructed through basic operations involving strictly smaller sets. Here, basic operations are:

• Products of sets
• Intersections of sets
• Unions of sets
• Functions between sets
• Subsets of sets

Then, you consider only the set of sets strictly smaller than $\kappa$, called a Grothendieck universe. Since virtually all set-theoretic constructions that occur in regular math can be boiled down to combinations of the basic operations, one never leaves this set. Having chosen a Grothendieck universe, one can then only consider categories whose objects are constructed in that Grothendieck universe. This means you get small categories, where you can genuinely talk about the set of objects and the set of morphisms.

The appeal of Grothendieck universes is that it makes the terminology simpler. The collection of all $R$-modules is a proper class, but only to preclude the existence of such pathological modules as 'the free $R$-module generated by the set of all sets which are not elements of themselves'. If you are actually working with $R$-modules, you will never accidentally try to create such a module; its just too ridiculously big. Therefore, it is sensible to consider some cut-off cardinality, past which all sets are 'too ridiculously big to matter'. Having made this cut-off, the majority of set-theoretic concerns disappear.

I should mention that this approach can be on shaky axiomatic foundations. The problem is that only one strongly inaccessible cardinal can be constructed in ZFC; the countable cardinal $\aleph_0$. This gives the Grothendieck universe of hereditary finite sets, and so this Grothendieck universe rules out the existence of any infinite set. Therefore, if you choose a bigger Grothendieck universe, you are implicitly including an assumption that a bigger strongly inaccessible cardinal exists. Personally, I am ok with this.

(Answer for 2.) One reason most mathematicians can avoid the theory of proper classes is through the use of Grothendieck universes. In brief, one chooses a strongly inaccessible cardinal $\kappa$, which is the cardinality of a set so big that it cannot be constructed through basic operations involving strictly smaller sets. Here, basic operations are:

• Products of sets
• Intersections of sets
• Unions of sets
• Functions between sets
• Subsets of sets

Then, you consider only the set of sets strictly smaller than $\kappa$, called a Grothendieck universe. Since virtually all set-theoretic constructions that occur in regular math can be boiled down to combinations of the basic operations, one never leaves this set. Having chosen a Grothendieck universe, one can then only consider categories whose objects are constructed in that Grothendieck universe. This means you get small categories, where you can genuinely talk about the set of objects and the set of morphisms.

The appeal of Grothendieck universes is that it makes the terminology simpler. The collection of all $R$-modules is a proper class, but only to preclude the existence of such pathological modules as 'the free $R$-module generated by the set of all sets which are not elements of themselves'. If you are actually working with $R$-modules, you will never accidentally try to create such a module; its just too ridiculously big. Therefore, it is sensible to consider some cut-off cardinality, past which all sets are 'too ridiculously big to matter'. Having made this cut-off, the majority of set-theoretic concerns disappear.

I should mention that this approach can be on shaky axiomatic foundations. The problem is that only one strongly inaccessible cardinal can be constructed in ZFC; the countable cardinal $\aleph_0$. This gives the Grothendieck universe of hereditary finite sets, and so this Grothendieck universe rules out the existence of any infinite set. Therefore, if you choose a bigger Grothendieck universe, you are implicitly including an assumption that a bigger strongly inaccessible cardinal exists. Personally, I am ok with this.