5. There is a variation of the generalized elements, called members in Mac Lane's book CWM. Namely, two morphisms $x,y$ with codomain $A$ are identified when there are epis $u,v$ such that $xu=yv$. Clearly this is symmetric and reflexive; in an abelian category it is also transitive. Mac Lane uses the notation $x \in_m A$. Then some properties of members are established, which are used to prove diagram lemmas in arbitrary abelian categories. In my opinion, this is far more efficient and enlightening than proving them via Freyd-Mitchell. And for me this is a perfect translation of the "element calculus" in abstract abelian category.

For other categories, you won't have a general answer what the best "element calculus" might be. It really depends on the context. I strongly agree with François G. Dorais' comment above. A better question would be: Given the problem xyz, how can I define elements in context abc in order to solve xyz?