It seems to me that you are trying to understand the set theoretical foundation of calculus. This would be equivalent to learning how to program in C++ and then insisting to learn how the CPU interprets the compiled code, and how the compiler works.
It is a useful knowledge, but not very useful for C++, or in this case -- for calculus.
If you wish to learn more about the interactions of set theory with other fields of mathematics, I suggest that first you get comfortable with following set theoretical related topics:
- Descriptive set theory,
- Basic topology,
- Cardinal arithmetic and basic PCF theory,
- Model theory.
Then you can apply these into measure theory, which is the modern extension of calculus; set theoretical topology; abstract algebra (many courses in advance model theory basically amount to algebra and algebraic geometry).
Studying these topics could take a couple of years, and by then you may find yourself interested in set theory per se. Let me give some basic recommendations for books.
- Moschovakis - Descriptive Set Theory.
- Engelking - General Topology.
- Holtz, Steffens, Weitz - Introduction to Cardinal Arithmetic.
- Chang, Keisler - Model Theory.