How does the forward speed of an auto-gyro relate to the rotation speed of the rotor during auto-rotation?
An autogyro flies at constant rotor RPMs. In vertical autorotation, that is a particular case, the rotor revs may be 10-15% less, but in straight and level flight, the rotor revs are higher, and always the same, independently of the speed of the gyro. It cant be otherwise, since, if the rotor revs went higher with the speed, the rotor thrust would also rise, and the gyro will gain altitude, no longer flying straight and level...

It's also true that if, flying s/l, the pilot opens the throttle, he also pushes the stick slightly forward in order to reduce the AoA of the disk, and thus keep flying s/l...