Is it permissible to shorten the prayer (qasr) throughout the week, given that it is a transitional period not exceeding ten months, and is it permissible to combine the Dhuhr and Asr prayers, either by advancing or delaying them, in the case of travel or temporary residency?

The days of your stay at your residence do not permit shortening (qasr) or combining (jam') prayers. However, for workdays, if they are less than four days (excluding the days of arrival and departure), you are considered a traveler.

If you are considered a traveler, it is permissible for you to shorten the four-rak'ah prayers and to combine Dhuhr with Asr and Maghrib with Isha, either by advancing them (taqdim) or delaying them (ta'khir). The combining of Dhuhr and Asr must take place after the time for Dhuhr has begun; it is not valid to combine them when departing in the morning. It is permissible to combine two prayers during travel within the time of one of them, according to the majority of scholars.