I have no idea whether there really was a tawsing stool, but there was certainly not a mobile one. I have many Scottish contacts and they have always told me that the tawse was applied on the hand.
As for mixed PE lessons, these are not a myth. At my grammar school in the 70s we did , from time to time , have mixed games lessons. These would consist of basketball, and volleyball, and I also remember playing rounders and hockey in mixed teams with the girls. This would only happen if the playing fields were out of order due to bad weather.We never had PE in the gymnasium with girls where they would have been dressed in the orange bloomers which I have mentioned before.
In one of the mixed hockey games I recall two boys being slippered for fighting, by the female games mistress. No doubt the girls enjoyed the sight.