How does wing aspect ratio influence lift and drag?
Your suspicion is leading you in the right direction. A lot of what you read on the Internet about induced drag is at least misleading if not blatantly wrong. But it is repeated by sloppy authors who don't ask the right questions.

If you have read stuff like "high AR creates stronger vortices at the tip of wings, which reduce lift", you hit such a completely wrong statement. The misleading version of it is high AR creates weaker vortices. For a hopefully better explanation, please turn to this answer.

Now to your questions:


  
  How does the aspect ratio influence induced drag, why is the AR in denominator?
  


The surprising answer: It is the lift per unit of span and the inverse of the square of flight speed which determine induced drag. Another influence is the distribution of lift over span, but aspect ratio does not play a role. See this excellent answer by @DeltaLima for an explanation, or this one for a longer one.


  
  How does it influence lift?
  


Indirectly, by increasing the lift curve slope.


  
  What really happens at the wing tips?
  


On the top side, suction turns the airflow inwards so the air gets a rotational component and some of it flows around the tip from bottom to top. Due to the small wingtip curvature, this vortex has low pressure at its core (and lets air condense early), but is overall weak and rather insignificant. The real trailing vortex results from a rollup of the wake, but even this does not create drag.


  
  What happens in the wind tunnel on a wing that spans its whole width?
  


Please read the answers to this question.