I followed the work in part (a) and agree with your conclusions. Well done! (And your presentation is easy to read)

Regarding part (b). Unfortunately your conclusion of, "Two parallel coincident planes and one non-parallel", is incorrect. Try plugging x=0 and y=0 into all three plane equations and you'll obtain three different values of z. In fact you can just see that the normals point in different directions so this statement can't be true.

If I was answering this question, I'd find the intersection of the planes in pairs {1,2} {1,3} {2,3}. This will give three lines of intersection (or perhaps you'll get the same line for all three). You've already found the intersection of planes 1&2. If the intersection of 1&3 is the same line as 1&2, then all three cross at the same line (no need to work out the intersection of 2&3).