The axiom schema of separation:
For all P and x, if P is a formula and x is a variable not free in P, then all closures of the following are axioms:
ExAy(y in x <-> (y in z & P))
So you see that the only technical matter about variables is that x is not free in the defining formula P. That is hardly "convoluted". Indeed it's straightforward and common sense. It's common sense that you wouldn't define a set, calling it 'x' with a condition that itself mentions x. Indeed convolultion (and contradiction) would result by allowing x to be free in P.
As to what predicates are allowable. This is rigorous and straightforward. Any formula P in which x does not occur free.
The axiom schema of separation expresses a straightforward notion: Given a set z, we can form the set x that is a subset of z and x has as members all and only those objects that are in z and that have property P. That is even common everyday reasoning. Given the set of all umbrellas we can also form the set of all umbrellas that are red. That is given the set z of all umbrellas, we have the set x that is a subset of z and such that the members of x are all and only those objects in z that are red.