Sorry, I don't understand your ... when teacher give answer he should be able to view his answer also based on courses and assignment.
This is what you can do using standard Moodle stuff:
Add a random glossary entry block and select your FAQ glossary
You can have it show a new FAQ item every day or every week or you can make it show only the last added item
In the footer a link is displayed Add your own entry!. You can change the text to something more specific. Clicking on the link will take students directly to a form to add a new glossary entry.
If you are willing to install code from contrib:
Jon made a questions module which looks perfectly fit for the task.