That's a great list! Being a grad student in mathematics, I find several of those way more upsetting than anything like Banach-Tarski or the hat problem in the linked post.
Oh, I agree. I'm totally comfortable with it, by now. I think the von-Neumann quote might apply here: "In mathematics, you don't understand things. You just get used to them."
Indeed, this is what intuition is. We often fail to notice where our childhood intuitions come from (experience), and so our adult desire to understand interferes with our ability to form new intuitions.
What's the difference? Uncountable sets does not respect out intuition about countable sets.
AC is independent from ZF, meaning it doesn't matter for any application to anything in the Universe, including physics or computing. Theory of uncountable objects is a game to play with symbols, the only thing "real" about them is that they are the firm boundary between the potentially reachable and the infinite void. The name itself conveys an important truth.
