To begin with, I am aware of these questions, which seems to be related: How do I fix someone's published error?, Examples of common false beliefs in mathematics, When have we lost a body of mathematics because errors were found?, etc...

My background: I am a senior undergraduate student in mathematics. Recently, I got a nice chance in a REU program, and started to read some journal articles. My impression was: any result in modern mathematics critically depends on another result, and that result depends on some other result, and ad infinitum.

On the other hand, some graduate students and professors in my university, who stand in quite intimate relations to me, say that, they do not check every details of proofs when they read mathematical monographs and research articles. They simply do not have enough time to read all the details and fill in the lines. (Clearly, I also do not read all the proofs in detail, if it seems to be so difficult or not much relevant to what I am interested in.)

Finally, I've been heard of some stories on fatal mathematical errors. To be honest, I do not understand what the errors precisely are. What I've been heard about are some "urban legends". (I intentionally didn't write down the details of these urban legends, since if I write down everything I've heard, maybe someone working in the mentioned field may feel insulted...)

For the above reasons, recently I am afraid of the situation where a field in mathematics collapse down because of a single, fatal, but very subtle error in the foundations of that field. In mathematics, everything seems to be so much intertwined, and it seems that no one actually checks every single detail in every mathematical articles.

But the mathematics community seems to be very sound. Maybe at least one of the followings are true:

Actually, a typical mathematical result does not depend that much on other results. So whenever if possible, a mathematician can check the details of every results which is of interest to him/her.

Strictly speaking, rigor is actually not that important. Even if a mathematical result turns out to be false, there is still something true in the statement. Therefore, only minor changes will be needed, and all the results depending on the turned-out-to-be-false result remains sound.

Here are my questions.

Why the whole mathematics remains so sound, even though humans are imperfect and quite often produce errors? Are my explanations above correct?

If a theorem turns out to be wrong, then mathematicians will try to correct (if possible) all the results depending on that theorem. How hard is this job? Isn't it very tedious and frustrating? I want to hear some personal stories.

As an undergraduate student, I want to know if anybody who is much wiser, older, or experienced, had the same fear as mine. (Again, I want to hear some personal stories.)

As an undergraduate student who will get into a graduate school in the near future, I want to get some advice. Should I stop worrying and believe the authors of the books and articles I read? When should I check all the details, and when should I just accept the theorem as given?

Thanks to everyone for reading my question.