As indicated by user337830, understanding your questions, hence their answers, depends on the foundational setting you use for mathematics.
Beyond a few remarks in MacLane's *Category Theory for the Working Mathematician* (Chapter I, Section 6) and the initial setting of Kashiwara/Schapira's *Categories and sheaves*, the standard litterature is rather poor on this matter, possibly reflecting the absence of interest of “working mathematicians” about foundations. Grothendieck's SGA 4 gave slightly more comments in Expose 1 (2 sections on universes, and an Appendix by Bourbaki). More positively, this deficiency may reflect the fact that for “usual mathematicas”, the precise foundational setting is rather innocuous.

For example, Kashiwara and Schapira introduce universes, but forget them right away, as does Grothendieck. (In K-S's book, this leads to unprecise expressions, such as “for a sufficiently large universe”, and some change of universes in the middle of a proof I can't understand the meaning of...)

The most naive point of view consists of having all categories and functors viewed as formulas in the language of set theory. It is OK for a lot of things such as Grothendieck's Galois equivalence of categories (but not for categories of functors, I would guess).

Then come universes or classes. The point of view of universes has the advantage of lying within classical (ZFC) set theory, while introducing classes requires another foundational system, Gödel-Bernays's (GB), say, something mathematicians may be reluctant to do.

As MacLane writes: if you have an universe $U$, you can call an element of $U$ a small set, and a subset of $U$ a class (or a large set), so that a model of ZFCU furnishes a model of GB.

For more complicated constructions, you'll need higher classes, or a (possibly infinite) sequence of nested universes. For example, categories of functors (categories of presheaves, for example) depend on the chosen universes; the fact that they are (or not) locally small is a theorem. I try never to forget that the fpqc-sheaf associated with a presheaf depends on the chosen universe (an example is given in Bosch/Lütkebohmert/Raynaud's book, *Néron models*).

Two papers sum up this with great clarity, I believe, and explain the merits/demerits of various approaches. One is due to Andreas Blass, *The interaction between category theory and set theory*, Contemp. Math. **30** (1984). I particularly like the (not so) recent preprint of Mike Shulman's, *Set theory for category theory* (2008, why is it still unpublished?).

In any case, I would advise you (if I ever may) to choose (possibly secretly) one foundational system you're OK with and to fix your terminology in consequence.