Is it permissible for a teacher to pose a question during class and award an extra mark on the exam to whoever answers it?

The reference for determining a teacher's actions is the educational system; if the system permits a certain action, then there is no harm in it. Otherwise, it is not permissible. Ibn 'Uthaymeen issued a fatwa stating that an individual teacher's action that violates the system is not permissible. However, if the action is by someone authorized to do so (such as increasing grades for students enrolled in Qur'an circles) and it is for encouraging good, then there is no harm. Similarly, if the system allows for good manners and etiquette to contribute to the mid-term grades, then the teacher is permitted to reduce the grades of those with bad manners and increase the grades of those with good manners, because he is entrusted with this and is accountable before Allah.