Replacement for uknown broken transistor - bipolar or MOSFET?
Your diode test results and operating waveforms indicate that it is an NPN bipolar transistor (if it was a FET the Gate drive voltage would be higher, and the Gate would measure open circuit to both Source and Drain). 

The circuit probably looks something like this:-







simulate this circuit &ndash; Schematic created using CircuitLab

The transistor should not run hot because both input and output currents are limited by R1 and R2. So there be a short a short somewhere, either in a component or between tracks on the PCB. The most likely culprit is FET1, which may have burned out and is now a short circuit from Drain to Gate and open circuit from Drain to Source. If this happens then Q1 will try to drive the motor directly, but won't be able to because it can't supply enough current.

Remove the FET from the board and measure resistance between Gate and Drain. It should be infinite. If it reads low resistance then the FET is shorted. You should also test the Schottky flyback diode (D1). If this shorts out the FET usually follows.