Can I use a transistor and a relay in conjunction to control the speed and directional rotation of a brushed motor?
Yes, you can use a relay for direction reversing of a brushed DC motor. It would use a DPDT relay as shown in the diagram below. And yes, at the same time you can use an N-Channel MOSFET transistor to control the speed of the motor by driving the MOSFET with a high speed PWM signal. The duty cycle of the PWM would control the motor speed. 



With the proper selection of MOSFET you would be able to drive the MOSFET from the PWM output from your Arduino. The direction relay coil is shown as using the same voltage as that used to power the motor. If the relay coil was a different voltage it could be connected to a separate voltage rail.