What are the rulings on shortening prayers (Salat al-Qasr) for a traveler who moves between two cities, and does the shortening apply to him in the country where he temporarily resides for two to three months, with repeated returns to his original city?

If you intend to reside in a country for four days or more, you must complete the four-rak'ah prayer, and it is not permissible to shorten it after arrival. As for being on the road, you may shorten and combine prayers if the distance reaches the distance for shortening prayers. If the duration of residence is less than four days, you may shorten the four-rak'ah prayer (Dhuhr, Asr, and Isha to two rak'ahs) provided the distance reached is approximately 83 kilometers, and combining prayers is also permissible, though not combining them is more preferable.