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][1] 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.


[1]:https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-27/issue-3/Hausdorffs-Grundz%C3%BCge-der-Mengenlehre/bams/1183425494.full