1. That sentence is correct. "to the exam" means on the date and place your exam is, and "for identification" means that they must make sure your name and birthday are correct. When put together, "to the exam for identification" is saying that whomever is giving you the exam needs to confirm your identity, and has asked you to bring your passport on the exam date.
2. In sentence a, to have correct grammar you would need to identify who "she" is. For example, "My brother is a student," said Rebecca. Or, "My brother is a student," said my sister.
It is usually only the first time you are introducing someone that you would use sentence a like that. It is easier and more common to use sentence b after that.
I hope this helps.