How much mathematicians rely on other results? When mathematicians do research, do they think every detail so that they know exactly every step what it takes to achieve the result from axioms? Or is it possible to work as a researcher without knowing every detail? For example, is it enough to know that for example Wiles proved the FLT and Helfgott proved the weak Goldbach conjecture or should I work through the details if I will apply those results in some publications?
 A: This depends both on the domain and the individual researchers, but I would say many (or even most) people rely in their research on results of which they do not know every detail (or even ignore many details) of the proofs. 
And, even those that in principle know everything they use certainly do not actively think about every detail of results they (re)use when doing new things. 
You specifically mentioned FLT. This yields in some sense a good example. The key part here was to show that certain elliptic curves are modular (if you do not know what this means ignore it, it does not matter for the moment); keyword: Taniyama-Shimura conjecture. Somewhat latter this result was generalized avoiding a condition of semi-stability and now there is a nice result called the Modularity Theorem.
Various researchers use this result, the Modularity Theorem, to solve other diophantinve equations for example or still other applications without knowing its proof (in detail). 
Another prominent example is the Classification Theorem for Finite Simple Groups. There are many papers that rely on this classification theorem, while this result is famous for its extremely long proof that hardly anybody knows in full detail. 
And, Blackbox theorems gives many more examples of this type of results.
What is however very important is to at least fully understand the statement of the results, all conditions that are imposed, what the conclusions mean precisely and so on, in detail with precise defintions (not vague paraphrases). If not there is a huge risk of commiting erros.  
A: This all depends a bit on what you mean by "knowing all the details".
From my experience an knowledge about this, most researchers depend heavily on other people's results. And one can't know every single detail of the results that one use. We have to rely on the fact (hope) that when something is published, then other people have checked the details. That is not to say that articles don't contain typos and even greater inaccuracies. It is very common to find smaller mistakes in papers.
I think that one thing that makes a researcher good is his or her ability to use results without knowing even the background of those results. You are somehow able to distill what "you need" from a paper without having to know a lot about it. You of course have to be convinced that the results used actually apply to your situation. That can't be any uncertainty about the statements that you are using. And the more your know, the better. You might then, of course, run your ideas by someone who knows more about that area to check that you haven't misunderstood anything.
This is the same "problem" that a Ph.D. student can "run into" when starting to do research. There is this whole world out there, and you are somehow expected to absorb what you need without having to spend years learning all the details. It can be challenging to go from a graduate level class where all details are presented, to having to just "accept stuff". This is, in my opinion, where a good advisor is essential. 
All that said, you, of course, want to know the details of your own research. That is, you need (in my opinion) to know the details of how these results are used.
