Is it permissible to combine and shorten prayers when traveling between one's governorate and place of study (more than 80 km), and is it obligatory to re-perform past prayers if that was a mistake?

It is permissible for a traveler to shorten the four-rak'ah prayers (qasr) and combine the common prayers (Dhuhr with Asr, and Maghrib with Ishaa) either by performing them earlier (jama' taqdim) or later (jama' ta'kheer), provided that their journey is lawful and its distance is not less than 83 kilometers after exceeding the houses of their village.

If the university is 83 km or more away from your village, you are permitted to shorten and combine prayers there and on the way to and from it. This ruling ceases upon your arrival back at the village.

However, if the distance is less than that, you are not permitted to shorten the prayers, and you must perform them in full. In this case, if you combined Dhuhr with Asr, your combining is valid, but shortening is not permissible. You must make up for all four-rak'ah prayers that you mistakenly shortened, as well as the prayers you shortened if the distance was less than the permissible distance for shortening or if it was within the city.

If you do not know the number of prayers you mistakenly shortened, continue to make them up until you are reasonably certain that your obligation has been fulfilled.