Context: Newton personally believed in the concept of absolute space since it reinforced the idea of an absolute God, but the entire premise was proven false by what we know as 'relativity of motion' which makes use of Newton's laws of motion.
Excerpts from 'A Brief History of Time' by Stephen Hawking:
"... Aristotle believed in a preferred state of rest, which any body would take up if it were not driven by some force or impulse. In particular, he thought that the earth was at rest. But it follows from Newton's laws that there is no unique standard of rest."
"Newton was very worried by this lack of absolute position, or absolute space, as it was called, because it did not accord with his idea of an absolute God. In fact, he refused to accept lack of absolute space, even though it was implied by his laws."
I'd say that depends on the specific god claim. Some of them are unfalsifiable, but others could be demonstrated to be true or false. I feel like your examples support this, because you could easily demonstrate the existence of a unicorn or a dragon by finding and capturing one. Also, the fact that they're fairly large land animals and there's zero confirmed sightings, no photographs and no captures is pretty good evidence that they likely don't actually exist (on Earth).
Generally, if the claim is that the god intervenes in some form, it should be testable. You could, for example, test whether prayer works. But if it's more of a deist god that supposedly just created the universe and then fucked off then that's probably not testable. But I don't think any of the most popular religions propose such a god and it also wouldn't really have any implications for human lives, so the claim isn't as interesting to me.
I don't like this argument very much because defining omnipotence in a way that includes logical impossibilities doesn't really make sense to me to begin with. I think it's more reasonable to define omnipotence as "can do anything that's possible".
I do however agree with the sentiment that we can't know anything with absolute certainty.
Yes, the history of science is refuting claims about God or gods. What is left is the untestable vagueness of the God of the Gaps.