Is it permissible for me to shorten and combine prayers upon returning home due to time constraints at the university and the difficulty of performing them on time, and can they be performed while sitting?

It is not permissible for a resident to shorten the four-rak'ah prayers or to combine prayers except out of necessity (such as rain or illness). Furthermore, standing in obligatory prayers is a pillar without which the prayer is not valid, except for the incapable. Therefore, what you mentioned about combining the four prayers and sitting during an obligatory prayer while being able to stand is not permissible. It is obligatory to perform each prayer at its appointed time, and prayer can be offered anywhere on campus. It is permissible to combine two prayers that share a time when there is difficulty in not combining them.