What stops a university from doing whatever they want when an academic misconduct allegation arises? I got permission from a professor to submit an essay to him that I wrote for one of his previous classes. It is academic misconduct to do this unless you receive permission from the professor. 

Say the professor changes his mind for whatever reason and I have his permission in writing, what is to stop him from going back on his word during or after the class? I understand the Dean and Department Chair would also be involved. If I prove that I had permission, what can stop the university from failing me on the class or assignment and/or doing whatever they want essentially since it's basically a private business/entity? Would I need to sue the school?
What's to stop him?  His ethical considerations, and the paper trail that was created.

If you prove you had permission, what's to stop the university from doing whatever they want?

Your university will have policies in place, and procedures for filing internal complaints.  Your energetic self-defense, using your paper trail and university policies and procedures, would be what would stop the university from doing whatever they want.

You might also be able to file a complaint with your state education department.

I don't know anything about lawsuits, except that they can take a long time.  Perhaps it would help to post at Law SE.