hey ! i,m not a helicopter specialist, but i think the advantages are the same like in contra rotaing propellers in aircraft.
the advantage is that on one hand you can transmit more power/thrust/lift for a given rotor diameter and on the other hand there are no power losses from a tangential airflow which is "wasted" on a single prop installation.
on the other hand its a good question why only kamov went for it - when it would be a real advantage for helicopters many others would design it the same way.
and i cannot say anything about stability differences .