What are the ramifications of introducing a universal set this way?

Question

What are the ramifications of introducing a universal set using this axiom? $$\exists x : \forall y (y\neq x \rightarrow y\in x)$$

Answer 1

Then $x \cup \{ x \}$ is the set of all sets and the usual paradoxes ensue.

Answer 2

The paradoxes of set theory don't follow from the existence of a universal set. Instead they follow from the combination of naive-like separation axioms, and a universal set.

To clarify, Russell's paradox showed that the Comprehension schema is inconsistent. Namely, if $\varphi(x)$ is a formula, then $\{x\mid\varphi(x)\}$ doesn't have to be a set. We then replace that by Bounded Comprehension, also known as Separation or Subset, stating that if $A$ is already a given set, and $\varphi(x)$ is a formula, then $\{a\in A\mid \varphi(a)\}$ is in fact a set.

While this axiom schema conflicts with the notion of a universal set, in many ways. We can instead choose to limit the formulas which define a subset like that. This is what Quine did in his set theory, New Foundations. But this makes set theory hard to work with, and we don't quite know what is the consistency strength of this set theory (although it's not expected to exceed that of $\sf ZF$, and even below that).

So a universal set is consistent, but not by "adding it to the existing axioms", but rather reshaping the entire axioms of set theory to accommodate such a set.