I think you're right that the OP's characterization of science is naive (though I think the everything-is-open idea is always there as an ideal, and I think that really matters), but all your examples here seem really dubious.

At 300 BCE they were not centrally concerned with proving via experiment that the earths is round. They instead cared about things making sense intuitively. The idea that it's important to prove claims via experiment came with Descartes into the scientific mosaic which happened much later.

Then how did it come about that Eratosthenes proved via experiment that the earth is round and estimated its radius? (A little later than 300 BCE, I think, but in the right ballpark.)

The idea that observation trumps theory was not unknown to the ancient Greeks.

"How does action at a distance work?" [...] became again an open question when Newton was shown to be right by the expedition that measured the shape of the earth.

Newton's work was recognized as first-rank science from the beginning, so the idea of action at a distance was taken seriously (at least) from the publication of the Principia. I don't know what expedition you are thinking of, but I am pretty sure it's nowhere near the truth to say that action at a distance was thought to be rubbish until an expedition measured the shape of the earth and thereby proved that Newton was right after all.

[EDITED to add: OK, so I guess you mean the French Geodesic Mission of 1735, which found that the earth is bulgy in the middle (as Newton said it should be) rather than elongated towards the poles (as Descartes claimed, on the basis of what we would now regard as a crackpot theory of vortices). That's hardly conclusive evidence for action at a distance, nor would the opposite result have been anything like conclusive the other way. But, anyway, a large part of the point of that expedition was that "Newton or Descartes?" was an open question. It would be nearer the truth to say that the expedition closed the question. But not much nearer, because as I say the question the survey resolved was not the question of action at a distance.]

the central dogma of molecular biology was considered a close question for a long time

The central dogma, at least as Crick stated it, says that the transcription from nucleotide sequence to amino acid sequence never goes the other way, with proteins getting decoded back into nucleic acids. So far as I know, this is still a closed question. Was it ever thought absolutely certain that there's no mechanism by which proteins can affect the information in nucleic acids?

Biologists were confident about the fact that a lot of the DNA is junk DNA that doesn't do anything.

They still are, and so far as I know they never claimed that all non-coding DNA is completely functionless. The "biologists thought most DNA was pure junk but now they've been stunned to find that some of it is useful" narrative is, I think, mostly hype.