Is it permissible to shorten the prayer when intending to reside for a month in a foreign country, and how then is congregational prayer performed in the mosque?

It is permissible for a traveler on a lawful journey, covering the distance of qasr (shortening prayers) (approximately 83 km), to avail themselves of the dispensations of travel, such as shortening prayers and breaking the fast in Ramadan, unless they intend to reside for four days or more. If they intend to reside for four days or more, or intend to reside for a month as in the question, then it is not permissible for them to shorten prayers or break the fast, and they are subject to the rulings of a resident. Even if shortening prayers is permissible for them, it is a condition that they do not pray behind someone who is completing their prayers; otherwise, they must complete the prayer.