Your third example is often cited as an unintuitive result, so I wouldn't mind getting rid of it. The first and second are easy enough to consider collateral damage, which we already have plenty of in basic math. But the fourth one is harder to give up. What would it look like without the axiom?
* Let X and Y be sets. Then either they have the same cardinality, or one is smaller than the other.
* Let X be an infinite set. Then there is a bijection between X and the cartesian product of X with itself, X × X.
* Tychonoff's theorem: every product of compact topological spaces is compact.