1. It's not. The work on "category-theoretic foundations," say HoTT, is (sociologically speaking) a highly niche topic. Most professional mathematicians who work on the foundations of mathematics do so in the framework of set theory.
2. ZFC is fully adequate for all the mathematics 99% of professional mathematicians do (and probably more than adequate in terms of strength). Also, all the mathematics 99% of mathematicians do is insensitive to foundational issues, so if you swapped ZFC for another axiomatization of similar strength, no one would notice.
3. I don't believe ZFC was the first to avoid the paradox, but yes, it does not suffer from Russell's Paradox.
2. ZFC is fully adequate for all the mathematics 99% of professional mathematicians do (and probably more than adequate in terms of strength). Also, all the mathematics 99% of mathematicians do is insensitive to foundational issues, so if you swapped ZFC for another axiomatization of similar strength, no one would notice.
3. I don't believe ZFC was the first to avoid the paradox, but yes, it does not suffer from Russell's Paradox.