What to say when you ask someone to to something?

So first of all your first sentence is incorrect, it should be, “I'm studying English and have learned some phrases you guys use when asking someone to do something. But I'm still confused which one should be used and where.”

 Would be proper. 

Every statement you wrote we use in any situation. You can really use any of the phrases you used to anyone in any situation & it is proper English & we would fully understand what you’re saying. 

 “Would you like to go out to dinner?” 

  “Can you pick that up for me please?”

With your friends, "Could you do me a favor and...?".

With your teachers, "Could I ask you a question?". 

Any of the above (which you mentioned) would be OK - except "will you", which would be very awkward. Just don't say, "will you...".

Colloquial phrasings:

"You want to close the window?" (Please do it immediately).

"You've got to speak up, I can't hear you!" (But don't shout).