What is the correct saying upon hearing the muezzin say "qad qāmati al-ṣalāh" (the prayer has begun)? Is it "aqāmahā Allāhu wa adāmahā" (may Allah establish it and perpetuate it) or repeating "qad qāmati al-ṣalāh"? And what are the hadiths indicating this, along with mentioning their narrators and authenticity?

It is prescribed for the listener of the Iqamah to repeat the words after the one calling it. There is no harm in saying: "May Allah establish it and perpetuate it (Aqamahallaahu wa Adamaha)" when the one calling the Iqamah says: "The prayer has been established (Qad Qamatis-Salah)," because a hadith to that effect is reported in Sunan Abi Dawud. Although the hadith is weak, many jurists permit acting upon it in matters concerning the virtues of deeds. Sheikh Ibn Uthaymeen mentioned that whoever says it should not be reproached, and whoever leaves it should not be reproached. On the other hand, some scholars believe that the one responding should say: "The prayer has been established (Qad Qamatis-Salah)" exactly as the one calling the Iqamah says, citing as evidence the hadith: "When you hear the Mu'adhdhin, say as he says."