We usually say IN (the) exams. Examples: I think I did well in my exams. In the exams I was nervous. In exams I am always nervous. However, we sometimes change it to ON if we are referring to specific parts of the exam. How did you do on the chemistry theory paper? (You could, of course, use IN)

In American English both are acceptable. In my experience, "on" is more common here.

You can say "in the exam". It is correct.