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forcing, large cardinals, descriptive set theory, infinite combinatorics, cardinal characteristics, forcing axioms, ultrapowers, measures, reflection, pcf theory, models of set theory, axioms of set theory, independence, axiom of choice, continuum hypothesis, determinacy, Borel equivalence relations, Boolean-valued models, embeddings, orders, relations, transfinite recursion, set theory as a foundation of mathematics, the philosophy of set theory.
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Understanding Specker's disproof of the axiom of choice in New Foundations
I think the answer (to your first question) is "yes". I'm not very familiar with New Foundations, but I can describe how it works in ZF(C), and hopefully this will give some insight.
So, if you are …