The primary reason the rotor disk follows the ship deck is because the swashplate stays aligned to the deck, and the rotor disk follows the swashplate. Disconnect the pitch links and the rotor doesn't follow the ship deck nearly as readily (hinge offset and other extraneous forces like elastomeric damping-friction-springiness are the cause for the alignment).
Or you could just say the rotor is not rigid in space - if you were to constrain a gyro in the way you suggest, secondary precesssion would occur but this doesn't happen either.
This does not lead to a sideways tilt of the rotor disk (or rotor cone), but this leads to a forward tilt. This means that the tilt hase 90 degree phase lag. This large scale phenomenon is called the gyroscopic effect.
No, this is called flapback.
But the blade is still attached to a center point so the centripetal force has a downward component relative to the original plane of rotation. This downward component leads to the blade decelerating towards its highest point much earlier than from the external forces, and then moving downward. This downward component of the centripetal force is proportional to the rotation speed, the flapping displacement and the mass of the blade.
No, it is aerodynamic damping - as the blade flaps up it sees a reduction in AoA and thus a reduction in CL.
an you explain in more detail, why you think that pure aerodynamic forces applied as a sinusoidal function of time would not lead to 180 degrees phase shift?
sorry, I misread your post - it is 180 degrees because the pitch change starts 90 degrees before max rate of pitch increase and hence max rate of flap up.