I do not have any real experience on this set-up, but am wondering if the black boxes you speak of are removable and if it is possible to switch places among the four? I think the secondary thing on the back of the motor is your encoder/counter. It sounds as though you have a motor locking up, possibly in the gearhead or in the encoder (I have cracked some encoder discs before from hitting them). If it has no heat when it stops, it could be tripped and shut down for a moment, hence the cooling. Try to dry run it without the gearhead attached to see if it still heats up and viseverse with the encoder.
I am sure you have thought of about everything, these problems can drive one nuts.