Am I permitted to receive financial compensation for work I do under the title "Teacher and Research Assistant," even though it is not stipulated in the terms of this title, and other students perform it without pay?

If the university specifies the tasks for which it provides funding, then these must be adhered to, based on the Prophet's (peace be upon him) saying: "Muslims are bound by their conditions." If your work does not meet the conditions for financial entitlement, then you are not permitted to take it except by informing the granting authority. If they agree to disburse it to you, then there is no harm; otherwise, not.