Reference for graduate-level text or monograph with focus on "the continuum"

I like your idea of such a course a lot! If it is appropriate to recommend a book in German language, I think this one could be the perfect match:

Oliver Deiser (2007): Reelle Zahlen: Das klassische Kontinuum und die natürlichen Folgen

I own this book and can say it covers all the topics that you mentioned, and it is certainly a graduate-level text. I find the quality of the exposition outstanding, and the range of topics quite unique. It covers the historical development quite extensively and gives many references, with a focus on the original sources.

I think there is also a 2nd edition from 2008.

A great textbook for your course would be "The Structure of the Real Line" by Lev Bukovský. It covers all of the topics you mentioned, except for the Banach-Tarski Paradox, and provides all necessary topological and measure-theoretic background.

For english references:

An history of mathematics book, but extremely well written and mathematically sophisticated, with tons of references (that might be useuful) that adress all such things is

• G. H. Moore's Zermelo's Axiom of Choice: Its Origins, Development, and Influence.

This should be my top pic. Other sources I know of:

• D. L. Cohn's Measure Theory.

has a very nice introduction to Polish spaces and analytic sets, and

• Folland's Real Analysis: Modern Techniques and Their Applications.

discusses the measure problem and Banach-Tarksi's Theorem, and has plenty references.

However, the level is more of upper-undergraduate then graduate, I think.

For descriptive set theory we have Krechis Classical descriptive set theory and Moschovakis Descriptive Set theory, but I guess that, by what you've said you know their content already.