Since the pulleys are not massless, they will not have zero accelerations. Also, the tension in a thread is the same throughout, so the tensions T_{1} and T_{2} are equal, and T_{3} and T_{4} are equal. What I meant in the previous post was that the tension of the thread connecting the pulleys will not be equal to the tensions of the threads below them. Write Newton's laws for the pulleys in terms of the tensions. Then write Newton's laws for every mass, keeping in mind that the masses accelerate along with the pulleys they are connected to.

Why is the tension in a thread the same throughout? If the pulley has a mass, a resultant torque is required to set it in rotational motion, thus the tension on one side has to be greater than the other side, no?

That is correct, but you have no information of the radius of the pulleys, which is required to write a complete equation for torque. If you "ignore" the rotation of the pulley, you do end up with the answer you mentioned in your post. I'm not sure why you can do this, but I guess the two lower pulleys are not free to rotate about their central axes, so only the threads slide across them without friction (thus having the same tension)

Is this a flaw in the question? Because when it says "pulley", it's only natural to assume that rotational motion is involved, rather than frictionless sliding.