suppose the motor exerts a torque on the rod when it spins.
So the rod rotates around its long axis (like a driveshaft)?
Is this torque independent of the length of the rod?
Mostly, yes. The torque will depend on the motor (it may be incapable of delivering torque above a certain value), and the load on the rod (if there is nothing on the other end, it will spin rapidly instead of develop a large torque. It would be similar to trying to push a snowflake with a force of 1N. You can't do that with your arm.
torque equation does not appear to be distance dependent.
If you just deal with torques as torques (as you would with a motor), there is no need to worry about the distance. But if you want to decompose torques into explicit forces, you have to take into account where they act.
But that is the distance from the axis of rotation. You asked about the length of the rod. A rod could be very long, yet all the forces on it are very near the axis of rotation. The length doesn't matter. But if you try to twist a screwdriver while holding the handle (a cm or two away from the axis) or if you twist it from the shaft (a few mm away from the axis), you will find that you are able to develop much more torque due to that extra distance from the axis. It doesn't matter in this case if the screwdriver is 6cm long or 25cm long.