Is the extra money received by the doctor, as he is about to pass the final fellowship exam, considered unlawful, given that he was counted as having obtained the fellowship and received a higher wage?

The amount received from the center's administration is permissible (halal) because the administration was aware of the true situation at the time of contracting for the wage. This does not contradict the rule of contracting based on the certificate, because the condition for a valid lease (ijarah) is mutual consent (taraḍī) regarding the work and the wage, and that has been fulfilled.