How to tell right from wrong, good from evil, and just from unjust? How to define what is moral, what is not? Given a situation, what actions is neither animal-like nor machine-like, but human? To answer these questions, one must build a solid foundation in ethics with pure logic. Many books claim they are logical and foundational; but I really wish to begin with the tried-and-true titles, i.e., the James Munkres, the Walter Rudin, and the Dummit and Foote of ethics-textbooks that show me how to construct theories from the absolute scratch. I am new to philosophy, any guidance would be greatly appreciated.
There exists indeed a book developing a proof-based ethics. It is the classic
The work looks like a textbook from mathematics with definitions, axioms, theorems and proofs. Nevertheless for me it seems an oddity. Hence I am sceptical that a proof based ethics is the right literary format to develop or to teach ethics.
You might be interested in Kenneth Binmore's work. He is a mathematician/philosopher who has published a lot of research on game-theoretic foundations of ethics. See in particular the two-volume Game Theory and the Social Contract. Given that you referenced Munkres and Rudin in your question, you'll probably appreciate that Binmore also authored a two-volume text in analysis.
This was a fairly active field of research in economic theory in the 1980s and 1990s. For an introduction, see Kenneth Arrow's and Amartya Sen's work on social choice theory - they also adhere to a level of formal rigour that would likely be to your satisfaction.