What is the ruling on combining and shortening prayers in Oman, given that the questioner was born in Qatar and returns to Qatar during long holidays, and what is the opinion of the four schools of thought on this matter?

As long as you are residing in Oman for university studies for a period exceeding four days, you are not considered a traveler, and therefore, it is not permissible for you to combine or shorten prayers. That is prescribed for you only while traveling.