Why do we use groups and not GROUPS? When a group is defined, we ask that the elements belong to a set. If we allow them to belong to a proper Class we get a GROUP.
What is the advantage of working with groups? What properties do we lose when we work with GROUPS?
An example of the notation can be found here:
http://www.mathe2.uni-bayreuth.de/stoll/papers/games12.pdf
 A: For the same reasons why most of the time we only talk about sets and not proper classes in general. In vast majority of contexts, proper classes are not needed at all.
In practice, we're interested in small objects. Proper classes arise mostly when we want some kind of very universal object. But even then the universality is usually rather superficial – you can just think that the supposed proper class is actually a set in a proper extension of the universe, and its only universal for objects of small size. In that way, being a proper  class is pretty relative.
A different problem is that for proper classes we don't have many constructions and theorems that are commonly used, like equivalence relations, quotients (so no isomorphism theorems!, no presentations!), wellorderings. If we want to add those things we're basically doing the same as what I mentioned in the previous paragraph: we make the universe larger.
I'm simplifying a bit, there are some subtleties involved, and many of these obstacles can be circumvented using some clever tricks, but generally proper classes are not important, nor do they happen too often if you aren't actually looking for them. If you want really big objects, you'd be better off assuming there's a sufficiently large strongly inaccessible cardinal and working in a restricted universe. That makes for far less of an ontological headache, and doesn't really make you lose anything of interest, as far as I can tell.
