I will agree that you have obviously missed some basic principles in class, but I like how you have identified what principles you have missed and asked those specific questions. That is worth something, IMO. If I may attempt to contribute to answering these questions.
1) What is j? j is simply a quick way to designate "compound". In the context of this problem, j refers either to "heptane" or to "octane".
Pj is the vapor pressure of pure octane or heptane, and Pc,j is the critical pressure of octane or heptane.
2) Xj? Which xj are you referring to? Your problem defines two Xj's. In the Wagner equation, Xj does not refer to mole fraction, but refers to the quantity 1-T/Tc for the Wagner equation. In part 2, we are indeed using xj to refer to mole fraction. You will need to keep track of which xj you are referring to as you solve the problem.
3) As others have noted, T=320 is given as a starting point for solving for T at P=1 atm for each VLE point. You could just as easily have picked 300 or 350 or whatever you felt like. Since I know that, for an ideal solution (assumption made in the problem statement), the boiliing points of the mixtures will be between the boiling points of the pure components, I probably would have picked a starting point closer to the boiling points of octane and heptane, but it makes little difference.
Does that help answer your questions?