I am a little thrown off with this.
Can we express Russell’s paradox as "A dog chases all and only those dogs that don’t chase themselves"? What is a formal proof showing that it is impossible for such a dog to exist?
So the reason this isn’t paradoxical is that there’s nothing inherently wrong with that definite description; it just fails to refer. There is no such dog!
The Russell class results in a paradoxical outcome for any theory of predicative set definitions in that if the description itself is perfectly legitimate then the method of specifying sets by description is unsound. The Russell set is not itself the Russell paradox!
Does the dog chase itself?
Assume it does:
Then it doesn't chase itself because it is not not chasing itself. Contradiction.
Assume it doesn't:
Then it does chase itself because it is not chasing itself. Contradiction.
Both cases lead to a contradiction, so such a dog can't exist.
And yes, this is the same situation as Russel's paradox.
Replace dog with Set, replace chase with contains.
Many paradoxes boil down to this or a similar proof. The impact or point of Russels paradox however is that you can't use arbitrary properties to define sets. This is an insight that means that naive set theory leads to a contradiction (because it assumes just that), and lead to set theory being reformulated a few years later.
ZFC does not assume that, for every property, there is a set of all things satisfying that property. Rather, it asserts that given any set X, any subset of X definable using first-order logic exists. The object R discussed above cannot be constructed in this fashion, and is therefore not a ZFC set. In some extensions of ZFC, objects like R are called proper classes.