The power of the motor cooling fan is connected to different power source than the motor starter output power. Would be the starter VSD or soft? Why?

no connection of vsd or softstarter with the fan if its powred by other path .it shows that the morot speed variation will never effect on the fan speed to maintain the temp.

This actually depends on the motor operation and the fan load. If the fan load is too large, and the motor operates on variable speed, then operating the Fan on VSD will be more efficient, but therefore the control of the Fan's VSD shall rely on a feedback of motor temperature, and in all cases the operation of the motor must be interlocked with the fan's trip so that to prevent the motor run if the fan is in fault.

But of course, if this will require much more control, that shall be avoided if the fan load is small, then it shall be controlled ON/OFF through a contactor, but always interlocked with the motor

Both of vsd and soft starter are given smooth start for the fan, while I preferred the vsd as you can control the speed of fan through it where the soft starter is run the fan on full speed. So the vsd will save power more than soft starter

VSD control in Hz, and control speed

Fan shall be feed from normal supply (not vsd or soft starter) through a contactor. The contactor closing/opening is controlled by motor operation. 

it would be soft starter for the motor , no no usage for the VSD if the fn is connected to other source.