For #3 on the quiz how do you know which tension equation to use?

For this one, you want to write separate $T_1$ and $T_2$ on each side of the pulley, and then use the rotational dynamics equations to get enough equations to solve for the unknowns. Unlike the massless pulley case, when you have a massive rotating pulley (i.e., a pulley with a non-zero moment of inertia), you can't assume the tension is equal on each side-- the rotation is produced by a difference in tension force in each direction, resulting in a net torque on the pulley. (If the tension were equal at each point in the rope along the pulley, there'd be no net force at any point tangential to the pulley, and so no net torque and no rotation.)