Does a teacher sin if he dedicates a quarter of an hour of a 45-minute class to talk with students, given that he is ahead in the material, and what is the expiation for not giving the class due to fatigue?

The reference for determining a teacher's obligations is the educational authority or whomever it authorizes, and the prevailing custom within that authority. If the authority does not obligate the teacher to cover the entire class period with instruction, but rather to complete the curriculum, then there is no blame in what the teacher does, and vice-versa. In the event a teacher falls short of his obligations, he must inform the responsible authority so they may pardon him, or he should return a portion of his salary commensurate with the shortfall.