How to use Rudder during takeoff in single engine propeller aircraft? e.g. Cessna 172
Is this common in real life flying? Are actual pilots consistently
  able to align the nose wheel to the runway centreline during takeoff
  in such single engine propeller plane or am I being too idealistic?


In my experience flying primarily light singles its not terribly difficult to stay aligned on the centerline during takeoff. Even my first solo was nicely centerline. However the more powerful the aircraft the more the tendency and the more control/care may be required. 


  How is the operation of rudders in an actual aircraft like? Slow
  smooth gentle depression e.g. to the right pedal each time the
  aircraft swerve to the left or are the left and right legs constantly
  moving in quick succession to keep the aircraft aligned to the runway
  centreline?


All control inputs should be smooth and with care. I generally ease onto the right pedal as I apply throttle and ease off/back on to stay center line. 


  Generally, how much of a push is required on the pedals during
  takeoff? It seems to be very very slight 1cm touches to my rudder
  pedal on the simulator. Is that realistic?


This is what most home sims are worst at. Real light aircraft rudder pedals take a fair bit of force to work an have far more than a 1cm throw. I would add a curve to your response so that the initial throw of the pedals is less sensitive.