Your schools name and address is listed on your 1098T, this is the name and address you will enter on your tax return for the 1098-T.
The students name and address is also listed on there, but generally, unless they have officially changed their address to a campus address, this will not be the same as the school address.
The filer of a 1098T would only be the school, if the school is on the list of Eligible Education Institutions. If the school is not an eligible institution, they will not provide you with a 1098-T and you will not be able to claim any type of education credits.
A student does not fill out a 1098-T. They or their parents (if claimed as a dependent) will enter the 1098-T provided by the school on their personal tax return. If they qualify for an education credit form 8863 will be filled out through TurboTax and attached to the individual tax return.