Paradoxes don't exist in reality (they do in hypothetical situations), so there is indeed no paradox as you correctly observe. Instead, most people answer this wrongly, for some reason. And for some reason we call situations where this happens "a paradox". Though I agree that we shouldn't.

Edit, ok, there are things like "This statement is false.", but we should perhaps stick to "self-referential problems" with those.

I think paradoxes just exist in our theories, languages, and formal systems when we make flawed assumptions or create inconsistent frameworks. But physical reality itself just is what it is - no contradictions, just phenomena we sometimes struggle to describe accurately.

If contradictions (paradoxes) can exist, then anything becomes possible through the principle of "explosion in logic". From a contradiction, any statement can be "proven" true. The whole foundation of rational thought would be undermined. Right?

The Medical Test Paradox or what's that called do exist in the sense that when a test is positive for a rare disease we always run a second one.