This is the fourth of Tim Urban's series on Elon Musk, and this time it's about some reasoning processes that are made explicit, which LW readers should find very familiar. It's a potentially useful explicit model of how to make decisions for yourself.
http://waitbutwhy.com/2015/11/the-cook-and-the-chef-musks-secret-sauce.html
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.
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.
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, 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?
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.
If you have Einstein saying: "God doesn't play dice" Einstein wasn't following the ideal to be open about him playing dice.
Newton was accepted much sooner in the UK than on the continent. On the continent the expedition was important.
The idea that Newton was simply accepted everywhere when he published his book is popular science mythology. The field of History and Philosophy of Science interprets things differently and that's why it's worthwhile to read people like Kuhn.
The French Geodesic Mission.
The point of his action wasn't to prove that it's round but to estimate the radius. Aristoteles already had a firm idea of the earth being round and most of the people afterwards believed that the earth was round because of the arguments of Aristoteles and not because of Eratosthenes.
The dogma is that DNA get's transcripted into RNA and that RNA get's translated into proteins.
Proteins aren't the point. The issue are retroviruses who transcribe RNA into DNA and thus violate the dogma. Historically that makes the term "dogma" quite silly as a dogma is by definition closed.
Not all but most non-coding DNA. The argument for that was that bacteria with are subject to strong evolutionary pressures and have had a lot more generations to kick out junk have less DNA. If I remember right there's also a fish who has significantly less DNA than it's close relatives.
You have biologists making the public think that all human DNA got sequenced on the basis that the 8% that they haven't sequenced doesn't really count and is likely unimportant.
Today it's much more an open question than it was 20 years ago.