Do contra-rotating propellers have the same RPM?
Most if not all contra-rotating propeller systems are geared together so that both sections turn at the same speed.  This is done so that as well as allowing effective conversion of very high power into thrust, torque and P-factor are cancelled, making the aircraft easier to fly (especially in a single-engine or two-into-one installation).

This is why you see the same diameter propeller in front as in the rear -- if they were turning at different rates, the slower one could be larger diameter (a limitation on diameter is transonic conditions at the tips) -- but you always see the same diameter on front and rear.