Is it permissible to play the Holy Quran in all mosques with the intention of repelling affliction, and is listening to the Quran considered an act of worship?

Listening to the Quran is an act of worship, but not if it harms people by raising the volume and keeping it on continuously, or if it makes them uncomfortable to stop listening because they are busy, unlike personal recitation or listening at home. As for warding off tribulation, it is achieved through repentance, seeking forgiveness, steadfastness in obedience to Allah, seeking His help, and taking the necessary means.