Need help analyzing a circuit?
Q1: The elements following the op amp (Red Rectangle) are supposed to
  condition the signal (ac signal from the aux jack), however are they
  really necessary ? And what is their purpose ? The ones in the yellow
  box i think are for smoothing the blinking of the Leds in order not
  the blink instantly but add a little dimming effect


It functions like a half wave rectifier (only using the positive portion of the wave coming into it and the cap and resistor filter it. You may be able to drive the transistor directly but it might not give you the effect you want. The other problem with driving the transistor directly is you would at minimum need a resistor to separate the low impedance sink of the transistor from the feedback network of the op amp. (there transistor might sink too much current and eliminating the cap could affect your sound signals)


  Q2: Here in the circuit they use "Virtual Ground" for the Op Amps,
  however can they be powered by a single supply ?


Virtual grounds are typically used for cables and other situations when your ground cant be trusted (like battery powered applications) to force the ground to be 0V. In this case, its not really a virtual ground, it's a reference functioning at half of the supply voltage and probably exists to create a bias current (which is really weird). The best thing would be to try it with the virutal ground or without and see which is better for your application.