Is it permissible for a student who fears missing an important class to pray Dhuhr at home individually instead of going to the mosque? And is this considered a legitimate excuse?

The preponderant opinion is that congregational prayer is obligatory, and it is not permissible to miss it except for a legitimate excuse. You must perform the prayer in congregation and then go to school. It is more virtuous to perform it in the mosque. If you are unable to do so, and waiting for the mosque congregation or seeking another congregation would cause harm to your livelihood, then there is no blame upon you for praying alone at home. Sheikh Ibn 'Uthaymeen issued a fatwa (definition) permitting the abandonment of congregational prayer for one who would suffer harm by missing a class.