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.
There are a number of strands to your situation:
The normal fear of being "found out" after stepping up a grade in education
Over-awe at the academic work of others in your field
Learning to "read" papers written in academese
The usual morale cycles that all people in all vocations feel
The need to refresh, recall who we are and what we are trying to do in life
1A. Almost everyone (even the most cynical) starting graduate study has an initial fear of finding the going too much for them. This is a natural and even rational reaction to their new situation. You say that you're not a super smart person. Fine. (Same here!) But you got to be selected as someone adequate for graduate school by professors who have been through it. That means that your selectors thought you had what it takes to succeed here: the mind, the will, the interest, imagination, energy, stamina, perseverence and so on. Now, you have to start looking back at your own journey, appreciate your achievement and start believing a bit more in yourself. In the beginning it's going to feel like blind faith as you haven't been here in graduate school before. But that's just a new location. Math is still math, exploring is exploring, reasoning is reasoning and good honest work leads to progress in this environment as in all others before.
2A. Most of the work in your field will have been done by more experienced researchers: for that fact alone they should be on a higher level. But the best of ideas are essentially simply ones that apply existing principles to new observations/situations/problems, or at least uninvestigated ones. Of course, some efforts work out well, more moderately so and others lead to nowhere too useful. But you just have to keep trying to apply the best in your own nature to understand more of nature itself.
None of us can really think like someone else - unless it so happens that they look at the world much the same way as we do. So we always have to find approaches that are the 'natural' ones for us - regardless of how many eyebrows are raised by others commenting on our efforts. For this reason - and not for any reasons of vanity or conceit - we have to find our own approach and follow it at our own natural tempo. Finding one's own perspective on the research topic is really the most important aspect of graduate study, more important that success (i.e. showy results) or failure (not so showy results) in the research itself. So stop comparing yourself with others. Just get your head down and work on steadily in your own way.
3A. I don't know ANYONE who hasn't got frustrated at poorly written academic papers, especially those in the scientific field. When we finally get what the author is trying to say, we think: Christ - is that all they mean ? It could've been said so much more simply! And it so often could. Even in the old days, an academic's reputation depended a lot on the quantity of papers put out rather than the quality of each. With little new to say, those authors often tend to dwell on obscure but ultimately irrelevant side detail. Today's university researchers are under even more peer pressure so expect no less of head-scratching at their published work. There is also a lot of protectionism in academic life, i.e. published work intended to suggest to us how important and impressive the conclusions are without revealing anything about the true perspective that enabled such insights. Personally, I found this aspect of university life the most depressing. I hope your department has few of this ilk.
You must also realise that while many papers will be recommended to you at the outset of your research, only a small few of these will be vital. The rest will be like postscripts to the important ones or sometimes a sort of copycat paper from a 'rival' researcher in the same field. Some will be specious, some honest but wrong and some nonsense. You'll need to develop a method of processing papers efficiently before they are studies in depth. Most colleges provide a sort of intro-to-research type of course that should cover efficient paper analysis.
4A. However good we are at our jobs, there will be times when things will just happen awkwardly, sometimes even several things not working out as they might at the same time! That is just as true in research as in all other areas of life. We all know people whose first client was impossible to please or whose married life began with a lot of misfortune. The thing with random misfortune is to just settle down our emotions and then resume our work at a careful pace. Eventually things have to come right. Really bad days when everything seems to go wrong can happen. In these times it might be no harm to go for a few beers and a good night's rest.
5A. Most students became very cynical after the first few months in college. After 4 years of a primary degree, it might be better for your own emotional balance to spend a year or two in the real world of work before doing graduate school. You have decided to go "straight through" and that is your choice. But don't underestimate the effect of the years of effort on your morale and work-life balance. Of course, you want to do well for your own sake, perhaps for the sake of your supportive parents too. (If you have a good relationship with your parents, try to call them every so often - you'll feel better after it and as they know your nature best, they may be able to advise you best) But watch the needle on the morale fuel tank. Take time out periodically to refresh yourself. How to do this depends on your own needs and what ever activity or company will recenter you. But try to do it regularly as morale always tells in the end and since campus life only seems to cater to it in a very limited way.