Why (what and when) do we cite?
I recommend that you discuss these issues with an academic mentor who understands your discipline. You may also find it useful to visit the writing/tutoring center at your institution. 

That said, I've provided some short answers below. My answers are not intended to be complete because many of the questions that you are asking cannot be answered by people who are not experts in your discipline.


  Should I cite a paper (i.e. give credit) for ideas that are in its literature review, and that are totally not related to the core/added value of the paper?


Generally, no. If you encounter something that you want to cite in a literature review, track down the original source. This should not be difficult, assuming that the paper you are reading cited the original source properly.


  Should I cite every paper related to one field?


No. Unless you are defining "field" extremely narrowly, it will be neither possible nor desirable to cite everything.


  Can I cite only not-that-much-cited source and not classic ones?


There is no universal answer to this question. The answer depends on your purposes for the paper and the disciplinary expectations in your field. Talk to a mentor.


  If I am writing an assertion that is general (e.g. "Polar bears have mostly white fur"), should I justify it?(!)


You do not have to cite things that are considered common knowledge, but the question is this: What exactly gets to be considered common knowledge? There's no universal answer. It depends who you are writing for and what you are writing about. Here are some rules of thumb that should generally serve you well in American academic writing situations:


If you would expect any given high school graduate to know something (like the color of polar bear fur), then you probably don't have to cite it.
If you aren't sure whether you need to cite something, cite it, or at least mark it as a problem. You can always read back through the paper later with a mentor, colleague, or tutor who can help you to identify unnecessary citations.



  Which is the criteria to determine when an assertion need to be proven by a citation, and when it is admited?


Again, there are no universal criteria for this. You'll learn the criteria that writers in your discipline go by as you continue to read and study in your discipline.


  When I define a key term for my paper, should I always rely on past definitions?


In general, you should rely on past definitions if those definitions are (1) technical and specific and (2) an important issue in your paper. If you wish to break away from established definitions, you should explain why and make an argument for the new definition. Finally, a tip: It is practically never necessary to cite a dictionary definition in an academic paper.