O.K. Where to start? This is my first post to this forum so I'll try not to stray. First, A properly designed gyro is both easy to fly and very safe. However, the number of properly designed gyros is small. I'll first address the turbulence issue. The stability in turbulence is determined by the point at which the rotor thrust line passes the longitudinal axis. If the rotor thrust line passes behind the longitudinal C.G. when an updraft is encountered the nose of the aircraft will lower into the new relative wind. When a downdraft is encountered, just the opposite will happen. If the rotor thrust line passes in front of the longitutinal c.g. then an updraft will cause the nose to pitch up and a down draft will cause the nose to pitch down, this is an unstable condition. The next question should be, well how do you make the rotor thrust line pass behind the c.g.? By placing the propeller thrust line at or below the vertical C.G. and installing a horizontal stabilizer. The dangerous part of flying a machine that is not stable in pitch is when the propellor thrust line is above the vertical C.G. a SIGNIFICANT nose down pitching moment is created when the aircraft approaches zero g. The rotor itself is somewhat tolerant of low G conditions as long as the time period is not excessive (more than 2 seconds, exact number depends on many factors such as blade weight and airfoil.) Machines that have a high thrust line, low vertical C.G. and no horizontal stabilizer will pitch over severly as a result of propellor thrust when the aircraft is unloaded in turbulance. This is the classic power push over. This event can happen very quickly and even quick reflexes by an experienced pilot to reduce power may not be timely enough. A gyro with centerline thrust and an adequate horizontal stabilizer are not prone to power push over, gust induced oscilation, or pilot induced oscilation.
Keep in mind that this information is abreviated. You could write volumes about the stability of the gyroplane. A good place to gather additional information about gyros is the web site
http://www.rotorcraft.com they have a conference section with a couple of useful downloads. One is an excellent article written by Jean Forcade on stability of the gyroplane.
I am a gyroplane CFI in the U.S. and will be happy to answer any specific questions via email. I will be out of town intermittently and away from email for up to a week at the time. I am also the Chief Test Pilot for the CarterCopter and don't have internet access when I travel for those duties. I'll shut up for now and I apologize for the spelling. Not one of my strong points.