What is the ruling on using the school's internet for personal purposes, given ignorance of any financial cost associated with it, and what is the ruling on repaying this money if it is impossible to identify those responsible? Is it permissible to compensate this amount with tuition fees that the employer waived for other individuals? How should the amount due be estimated? Is it permissible to donate the amount to a poor country if the entitled party is wealthy?

If the school permits students to use the internet, then there is no harm. However, if it does not permit it, and the usage was extensive and incurred a significant cost, then you must repay the cost to the school. If you do not know the exact amount, you should estimate what you predominantly believe would cover it. This cost is not waived by expenses your mother may have paid. It is not permissible to repay the cost to another entity or another country unless you are unable to reach the rightful owners, in which case you should spend the money on matters that benefit Muslims. However, if the cost is customarily minor and overlooked, then you are not obliged to repay it.