Professor keeps students tests. No taking notes home when reviewing them. A friend of mine has a professor that does not allow the students to take home their graded test. The process the teacher has is like this:

Students come to class and the professor has everyone put everything away except a pencil. The tests are handed back and the students are allowed to see what they got wrong, and make notes on the test itself and ask questions. At the end of class, all tests are given back to the professor, and the students are not allowed to keep any notes made. Students may go to office hours to "visit" their test again if they would like. 

The online quizzes are also peculiar, as they only show if a student got a question right or wrong, leaving out both the correct answer and the answer the student gave. 

The reason the professor is so secretive about the tests is she allegedly reuses the tests. As such, she doesn't want students to be able to get last semester's test and be able to cheat from it. The professor thus far has been rather terse/short with students when they ask about how she runs the class. As such, only a couple students have asked her about these things.

My questions are:

Given the circumstances, is there any point in trying to ask the teacher to do things differently so that students can better learn from their mistakes?

Would talking to the department head be beneficial or would that be pointless and possibly rude?
Talking to anyone other than the professor them self comes off to me as pointless because there are typically no similarities between how each professor handles their course material.

A common reason in some of my courses was this: some questions come from a publisher's test bank, and the university has agreed that materials containing the questions won't be distributed. It doesn't make sense to me at all, and end up a way to continually reap money as banks change, ultimately never helping the students.

What I would do is schedule regular office visits if it's within her time to do so to go over the course material you're struggling with. My suspicion is she's rather set in her way. Yoou definitely don't want to go over her head as this would earn some demerits!