So to follow your logic, we can't even define a set properly, at least not completely.

By my logic we can't define that particuar set, because it doesn't exist.

Next is the hard part. Define R as the set of all sets not in S.

Is there any reason to assume that R will exist simply because S exists? Is there an axiom in set theory to the effect that "Every set must have a complement"? I'm not being glib or rhetorical in those two questions, and those questions are really the meat of the issue. There has to be a reason why R should exist for Russel's Paradox to truly be a paradox. I didn't notice the reason in the linked articles, and I'm curious.

Can something be both in the set and not in the set?

No it can't, and that's the proof that R doesn't exist.

So to follow your logic, we can't even define a set properly, at least not completely. We can only say what is in certain kinds of sets, and not what isn't in them.

This comes back to assuming that R should exist. There isn’t a problem with being unable to define non-existent sets. Heck, I don't define non-existent sets in my sleep! If there is a reason that R should exist, then we’ve got a problem.