Non planarity of K3,3
This explanation is wrong. In principle you could prove that $K_{3,3}$ is not planar by enumerating all types of planar embeddings, and show that some paths must cross. If I remember correctly this is the path chosen in some lecture notes, so if you look around you might be able to find this argument. Another route, taken by your lecture notes, is to use Euler's formula.

When your notes say that $K_{3,3}$ isn't planar since it "has a cycle", they were trying to provide you some intuition, not too successfully I reckon. The claim requires a proof which is indeed given later in the notes.