Is it obligatory to pay for the use of an unoccupied room in university housing, and is it necessary to remind those who were taught this method to pay for themselves, or is it permissible to pay on their behalf, and does the demand for payment drop if the housing official forgives it?

You must pay the room rent to those in charge of it, because benefiting from utilities is guaranteed in Islamic law according to the majority of scholars. Moreover, utilities are considered a form of wealth that must be guaranteed in cases of damage or usurpation. If those in charge allowed you to stay and were authorized to do so, then you are absolved. As for those whom you directed to that place, you are not liable for any guarantee; the guarantee is incumbent upon the direct beneficiary. However, you must advise them of the obligation to repent and compensate.