Is a student sinful if she doesn't cross out her answer due to her obsessive thoughts that she cheated after the teacher said, "May Allah not forgive whoever cheats"? And what should she do to get rid of these obsessive thoughts?

Cheating is forbidden, and the teacher's forgiveness does not change its ruling because it is a public right. One must be cautious of whispers (waswas) that might suggest cheating, especially since the doubts of one afflicted by waswas are often imaginary and lack reality. The cure for this is to turn away from these thoughts, strive diligently in studies, and make supplication.