Can helicopters do a full and free check of flight controls with engine(s) running before lifting off?
As Fareast says, the check is done before engine start. The controls are pretty stiff, with no hydraulics, and in some machines the controls won't move at all without hydraulics. Once the rotors are turning, a full pedal check would spin the aircraft on the ground and perhaps damage the tail boom attachments; full cyclic would hit the stops and perhaps chop off the tail; full collective sees you leap into the air.
Simple answer: no.