Is covering the head obligatory or Sunnah in prayer? And is wearing the ‘araqiyeh (skullcap) considered a dress of fame (libas al-shuhrah) that is prohibited, especially if it is used to distinguish between people in a Muslim society where only those who pray wear it?

Covering a man's head during prayer is a recommended act, not an obligatory one. If it is customary for the people of a certain country, then it is not considered a garment of ostentation (libas al-shuhrah), because a garment of ostentation is that which is worn to differ from people for the sake of distinction. The noble Hadith states: "Whoever wears a garment of ostentation, Allah will clothe him with a garment of humiliation on the Day of Resurrection."