I'll attempt an answer to question 1. Hausdorff was entitled to think that set theory was not yet mature, because his own 1914 book made considerable advances on what had been done previously (notably by Cantor and Zermelo). It is worth reading the glowing review in the 1920 Bulletin of the AMS to see how his book changed the perception of set theory by mathematicians. Just to mention two of his contributions: definition of a topological space, and the paradoxical decomposition of the sphere that paved the way for the Banach-Tarski paradox.

I'll attempt an answer to question 1. Hausdorff was entitled to think that set theory was not yet mature, because his own 1914 book made considerable advances on what had been done previously (notably by Cantor and Zermelo). It is worth reading the glowing review in the 1920 Bulletin of the AMS to see how his book changed the perception of set theory by mathematicians. Just to mention two of his contributions: definition of a topological space, and the paradoxical decomposition of the sphere that paved the way for the Banach-Tarski paradox.