Driving torque and load torque of a pmdc motor
My question is this: if the torque due to the weight = the torque due to the motor, shouldn't the weight be stationary? How does it move up? Obviously the torque due to the motor has to be greater than the torque due to the weight to lift it off the ground right?


No, it doesn't necessarily be stationary.
Consider the translational example of a car driving on the road. To accelerate, it needs a needs a remaining force. But when the force of the motor that pulls the car forwards and the friction forces that work against it, cancel each other out, the car will move with constant velocity.
Likewise, when the torque due to the weight = the torque due to the motor, the disc will rotate with constant angular velocity and the weight will therefore have a constant velocity as well.


  Moreover, how does the motor draw the right amount of current, thereby maintaining constant torque to keep the velocity constant? How does it not accelerate the weight by producing more torque than the load?


Your question is justified.
The motor does not know what torque it should 'produce' in order to cancel the torque caused by the gravitational force.
You need a controller that senses the position/velocity of the mass and will apply a voltage such that the motor torque cancels the torque caused by the gravitational force.