Is it obligatory to perform the five daily prayers in the mosque, and is it permissible to perform them at home or some of them during the examination period?

It is not permissible to perform prayers at home, even during examination days, because this is not considered a legitimate excuse to abandon congregational prayer. You can combine studying and performing prayer in congregation without harm, and congregational prayer may be a cause for dispelling distress regarding knowledge. Congregational prayer was not waived for Muslims even during fighting (Salat al-Khawf - the prayer of fear), nor was it waived for the blind.