Is my father entitled to demand the student pay the rent for the exam week, and is the amount paid in advance for the previous month, of which the student only attended one week, lawful?

The basic principle is that a rental (ijarah) contract is binding and cannot be rescinded except with the mutual consent of both parties. The hired person (ajir) is entitled to their wage as long as they are ready to work. However, some jurists, like the Hanafis, permit the rescission of an ijarah contract due to a sudden, unforeseen excuse (ʿudhr ṭāriʾ), an example of which is the illness of the lessee.

It is permissible to rescind a rental contract by mutual agreement of the parties, or due to a sudden, unforeseen excuse, or due to a defect that occurred in the rented item which impairs its usability.

Based on this, the student has the right to rescind the contract due to his illness and would pay the rent for the period he used the car. Otherwise, if he does not rescind the contract, he would be liable for the full month's rent.