Is using the paid internet provided by the university—which the university stipulates must be used for academic purposes and prohibits chat—for the purposes of Islamic daʿwah (proselytizing), issuing fatwas, and chatting, permissible or forbidden, and will the one who does so be rewarded, and what is the expiation for doing so if it is forbidden?

If the university prohibits the use of the internet for non-academic purposes, students must adhere to this, based on the Prophet's (PBUH) saying: "Muslims are bound by their conditions." The interpretation and scope of this condition should be referred to the authorized personnel at the university. If a student violates this, their expiation is to refrain from using the internet except for what the university has permitted.

Regarding your question about earning reward by someone embracing Islam through you, this is achieved if Allah accepts it from you, based on the Prophet's (PBUH) saying: "For Allah to guide one man through you is better for you than red camels." The fact that this occurred through the use of the internet without permission does not prevent the earning of reward, as the transgression here does not invalidate the good deed.