Is it permissible to perform all prayers, except for the Friday prayer, in the residence's prayer room without going to the mosque, and does one receive the same reward as praying in the mosque by doing so?

The fundamental principle is to consider the distance separating you from the mosque. If the mosque is so far that the adhan (call to prayer) cannot be heard from it by the unaided voice, then it is permissible for you to establish the congregational prayer in your prayer room (musalla). However, if you can hear the adhan, then it is obligatory for you to pray in the mosque, due to the Prophet's (peace be upon him) statement to the blind man: "Do you hear the call to prayer?" He said, "Yes." The Prophet said, "Then respond."

Since you do not have authority over the students, and as a way of encouraging them to pray and to attain the reward of congregational prayer, there is no harm in you praying with them in the musalla. This is better than them praying individually, and what that entails in terms of delay or lack of tranquility. The Messenger of Allah (peace be upon him) said: "A man's prayer in congregation is multiplied twenty-five times over his prayer in his house and in his marketplace." If possible, arrange for a weekly sermon for them in the musalla.