The topic you are interested is quite a large one. It includes the vast majority of the human condition, ranging from religion to politics to sciences to philosophy to mathematics. Hopefully you don't expect just one answer!
Your last question is, in my opinion, the most interesting. Indeed there are those who argue that we cannot know anything at all. These are known as skeptics, and there are a variety of categorizations of them based on how wide of a claim they make. I, myself, am strongly impacted by Agrippa the Skeptic and his 5 tropes, which were then rephrased many centuries later as the Münchhausen trilemma, which claims that any logical proof must rely upon one of three things:
- A circular argument
- A reduction to infinity.
- A unproven statement (an axiom)
Such an argument makes the strong case that one cannot define "knowledge" to be a provable thing if we seek to avoid all of those three things. And surely someone asking the question of "what can we know" is concerned with the oddities that arise if any one of those three appear.
Indeed, skeptics have the most curious problem one can come across. At the extreme, they state that you cannot know anything. Yet, to be consistent, they must not know that you cannot know anything, for such would be a contradiction. There are countless opinions about how to resolve this, which are certainly beyond the scope of the question.
One you accept that there are people who will make the claim that we cannot know anything, the rest of the questions become more interesting. They refuse to settle down, because there will always be that one unfinished corner of the argument. It encourages a pragmatic approach to the specific issues of politics and science that you address. People do indeed apply their beliefs over others (for whatever that phrase means), and people do claim authority. The validity of those claims is incredibly complicated, especially on a national level, but if I may simplify, I'd like to refer to an argument Alan Watts made. He was talking about the authority of the guru, and why a guru has authority. He made the argument that the guru has authority because the student gives him or her the authority. If the student did not give such authority, then the guru simply would have none. Watts makes a great argument, which I might be able to dig up if you were interested, but the intuitive sense of the phrasing does argue for itself. Our governments have authority because we citizens give them authority. This may be given freely. It may be given at birth before you know any better. Or it might even be given at gunpoint, when the alternative to ceeding authority is death. Whether we find such acts ethical or not, we must admit that it is possible to ceed authority in each case.
Science is another interesting situation. You say "its fairly easy to identify those that make correct claims, because correct claims must be e.g. repeatable." In theory, this is true. In practice, it is more nuanced. Consider the announcement of a new particle from the LHC. While in theory this is a repeatable experiment, in practice, there's no collider in town that can actually do it besides the LHC. When they make their claim, nobody else can repeat their experiment besides the LHC. As such, if you look at how groups of scientists at the LHC announce their discoveries, they spent extraordinary amounts of effort to address this issue. They re-run the experiments countless times, publish all sorts of low level data, and take care to announce only their observations. Even when they announced the "discovery" of the Higgs Boson, they took great care to phrase their statement based on statistics and observations. All because this issue of authority and knowledge is indeed an issue for science. They have to pay attention to it just like the politicians.