The first two "proofs" you mentioned as a reason to assume the concept of dark matter are not really proofs. They are just a nice way to explain otherwise flawed equations.
I agree. They consist of indirect evidence, like any evidence for dark matter that we have today. We still miss the direct proof (that is what we search so hard since many years). What I like with dark matter is that a single concept can explain many different problems. There is no other alternative that does this similarly well. However, until we find out, alternatives must be considered too, even if they do not look as good.
hat if gravity is just different than we believe, but without the presence of otherwise never observed additional matter?
This option is not ruled out. In terms of significance, it does not work as well as dark matter (but I insist, it is not excluded too). At the end, it is just my interests that lie on dark matter. You can find other physicists with other interests working on modified gravity theories.
Also I am aware that dark matter "explains" why the cosmos expands not linear, but increases it´s expansion rate. Again no proof in my opinion.
I guess you meant here dark energy (which I didn't discuss at all). We know even less on this one than for dark matter, and proofs are missing. So yes, you are correct. We have no clue what it is.