Can helicopters have engines running while rotor blade stays stationary?
Yes, many helicopters (but not all) can do this. There are 2 main methods: 


a clutch between engine and rotor system
for turbines only: a rotor brake on the engine output shaft. The engine contains a free turbine (i.e. a turbine stage that isn't attached to the compressor) that drives the rotor system. If you brake this shaft, the engine can run without spinning the rotors.