1. When I watch American or British dramas, there is a lot of slang. Slang is derived from their own distinctive cultures, then how do we learn that?
  To study slang in American dramas, choose one and watch it without any pausing and subtitles. Try to understand actors’ feelings, language sense with general situations while you watch it repeatedly. Do it again with English subtitles, and then try to speak and repeat the script as if you are an actor.
Slang usually comes and goes easily. If you want to keep it, be awake to research it on the internet.
Recommended site: urbandictionary.com

2. In Korean, it is clear to tell if the speech is respectful or not as there are honorific expressions. How can we tell the difference between respectful expressions and disrespectful ones in English?
  There may not be much difference in the use of nouns but the tone is decided by the use of verbs. Disrespectful expressions tend to be blunt and direct, somewhat imperative. In contrast, respectful expressions are oftentimes indirect or interrogative.

3. Many English words have different meanings in different situations. How can we choose the proper use in each situation?
  Above all, you need to develop your reading comprehension. Do not just read anything but try to understand it with an English-English dictionary (ex. Longman). I think English newspapers are one of the best ways to improve your comprehension as they have various reading materials in many fields, so you can read them. To assort suitable words or expressions, you need to be a student of wide and rich experience.