Is a female student's failure in her studies considered a tribulation from Allah, and does the mosque have a role in this failure, as her family believes, and is the niqab obligatory despite the family's opposition?

A drop in your exam grades might be a trial or due to your negligence in studying. Therefore, you should strive to be a good role model for a committed Muslim woman. It is not obligatory for a woman to attend the mosque for prayers; her prayer at home is better, and so is her studying. Make sure to befriend committed sisters. There is no harm in attending some lessons while balancing your studies and your parents' wishes, and give everyone their due rights.

Regarding the niqab, you can refer to Fatwa No. 32440.