What happens to the angular velocity and hence the angular momentum of the disk before, during, and after the application of a brief torque? Create...

If we apply a torque we will produce an angular acceleration. This follows from the fact that torque = moment of inertia x angular acceleration.

So before the torque there is no angular momentum of the disc. The disc does not rotate. As the torque is applied to the spool the spool experiences a change in its angular momentum. Since angular momentum of the system overall is conserved there is a corresponding change in the angular momentum of the disc. Torque is nothing more than the rate of change of angular momentum. So after the torque is applied the disc continues to spin. It continues in its state of rotational motion with the same final velocity it acquired when the torque is applied.

In rotational motion torque is analogous to force and moment of inertia is analogous to mass. Thus a first law of rotational motion is that a system in rotational motion will continue in its state of uniform angular motion unless acted upon by an external torque.