What is the authenticity of the Iqamah for prayer in the form mentioned in the question, and is it permissible to pray behind someone who performs it this way, and is there any legal evidence to prove this manner?

Imam Abu Hanifa, Al-Thawri, and Ibn Al-Mubarak were of the opinion that the Iqamah (definition) should be recited twice (i.e., its phrases repeated), based on the Hadith (definition) of Abu Mahdhurah. As for the majority of scholars, they hold that the Iqamah should not be recited twice. The matter is broad, falling under the category of "diversity in differences," meaning that both reciting it twice and reciting it once are permissible. Therefore, it is permissible to pray behind someone who recites the Iqamah twice, and one should not abandon the prayer for that reason.