I was told that at the time many mathematicians didn't really care about Russell's paradox. They believed that most sets you would encounter in everyday mathematics are too small to be pathological like that.
Much like nowadays many mathematicians don't really pay attention to large cardinal research for the same reason (and then there are those guys who just use them regardless).
The main implication from Russell's paradox is that not every definable collection makes a set. This is quite a shock to the naive approach taken at the end of the 19th century.
There are two major results (both, I believe, are due to von Neumann). The first is the development of the concept of a class, namely a definable collection. This meant that we can talk about nonexistent collections as long as we can describe them by a formula. This allowed in some sense "virtual sets" like the Russell class, or other paradoxes, to virtually exists as classes. We could refer to them, manipulate them, but they are not objects in the semantical universe (and so they don't really exist).
The second result is the idea of describing the universe as an increasing union of sets, i.e. the von Neumann hierarchy. This allows us to avoid the paradox in the sense that whenever we want to describe some collection, we can limit ourselves to a certain point in the hierarchy and see how the collection looks like at that point. In fact a definable collection is a class if and only if it is bounded in the hierarchy.
Outside of set theory, the Russell paradox doesn't have much impact, or rather we cannot see it nowadays when ZFC (a set theory in which the paradox is resolved) is the de facto meta-theory of mathematics. However you can see remnants of the paradox in every self-referential definition, like the category of all categories.
As for your last question, Russell's paradox (as most set theory) has little (or virtually no) relation to the physical world, as much as we know. Applied mathematics, everything physicists do, and real world-expressible shenanigans are usually very very small in terms of sets (I mean most are bounded below the power set of the real line).