Cessna 172 steering system and torque link purpose
The Cessna 172 (152, 175, 177, and 182 as well as others) all use steerable nose wheel by pushing (not depressing) the rudder pedals in the direction you want to go. If you want the nose to turn to the right, you push the right pedal forward, same with the left. You can augment this with differential steering if tighter turns are needed.

So there are actually 3 ways to steer the 172:


The nosewheel steering using rudder pedals. This puts pressure on the bungee system that turns the nose wheel left/right. Pushing the pedal doesn't directly move the wheel, you need to move the aircraft forward a little bit for the wheel to turn.
Differential braking can help make tight turns, this causes the nose wheel to "caster" or turn based on which direction you are attempting to turn the aircraft.
Some turning force is provided by the tail rudder, it isn't usually a lot but it is noticeable when landing.



  Is there any connections between rudder or toe brake and nose landing gear related to steering? What is the purpose of torque link in front oleo ?


There is no connection between the rudder or toe brake and the wheel (in the purest sense, the rudder pedals are also connected to the bungees for steering), the rudder pedals  move the bungee system that puts pressure on one side or the other of the torque link on the oleo strut. 


  Does nose wheel even turn ?


Yes! All tricycle gear aircraft with a nose wheel must allow the nose wheel to turn at least a little bit. The 172 allows the rudder pedals to give up to 10° of wheel movement while differential braking will make the wheel move up to 30°, this is outlined in Section 7 of the 172 POH (I won't link the entire POH here, but this is the excerpt):


Source: 172SP Pilots Operating Handbook


If you update your question with a picture of the fairing you are talking about I can comment on that, but really it should be a separate question.