Is it permissible to shorten prayers (qasr) in a house located 300 km away that is visited for three days weekly, and is it valid to pray Dhuhr instead of Jumu'ah prayer in this case?

Whoever travels a distance permitting Qasr (shortening prayers), which is 85 km, is allowed to shorten and combine prayers while en route.

Upon reaching their destination, if it is their place of residence, they complete the prayers. If it is not their place of residence—even if they own a house or have relatives there—they shorten prayers, unless they intend to reside there for more than four days.

This is supported by the actions of the Companions during the فتح مكة (Conquest of Mecca) and their Hajj. Some of them had homes and relatives in Mecca, yet they shortened their prayers, and the Prophet (peace be upon him) did not order them to pray in full.

As stated in the fatwas of the Permanent Committee (اللجنة الدائمة), for someone who has two homes in two different countries and resides in both, it is not permissible for them to avail themselves of travel concessions while staying in either of the two countries. However, it is permissible for them to do so while traveling between them, provided the distance is a distance permitting Qasr.