[sesqu]

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?

[ionfish]

Coming from a set-theoretic perspective, I suppose I've got so used to Tarski's theorem that I consider it intuitive.

As far as Tychonoff's theorem goes, you might find this paper interesting:

http://matwbn.icm.edu.pl/ksiazki/fm/fm113/fm11313.pdf