I'm a starting graduate student in math. Without a doubt I'm not a genius or even that smart, and that creates a serious problem: I often lose my nerve while doing research or studying other people's paper. I feel that some of those people (if not most) are just too good and I crack under the pressure to produce works of similar quality. I know the right thing to do is to be "brave" and power through and see where I end up. But this issue of "cowardice" has severely hurt my productivity. Do successful researchers have same problem or are they just totally engrossed with their work and do not care about anything else? How can I somehow be more disciplined with my emotions?
I think that you may experience what can be characterized as Frustration.
frustration is a common emotional response to opposition, related to anger, annoyance and disappointment.
As you're reading scientific papers, you're on higher level of scientific work than is expected from graduate student - I did not read too much papers during my MSc.
And it depends also what is expected of you by your colleagues, so compare whether your frustration is not combined with Imposter syndrome mentioned in comments.
E.g. Authors usually spent several months or years to produce some results and to write a paper. So understanding the paper may take also some time to readers. Other problem is that some papers are so brief thus it's rather a problem of the authors if they can't explain their scientific problem or solution to somebody outside their domain niche.
I have been experiencing frustration personally many times during my MSc, PhD and my current postdocs projects. When it happened I gave myself a time limit. So during that time limit it's good to consult the problem with somebody else. If I don't make any progress and frustration continues after the time limit, I give up and move to somewhere else.
The limit and tolerance to frustration is very subjective. Nobody should expect you to grasp all problems and in short time. My oppinion is that mental health is more important than scientific achievements.