Is it permissible to perform the Fajr prayer 7 minutes after the adhan at the beginning of its time, and is there a specific time between the adhan and the iqamah?

There is no specific time limit between the Adhan and Iqamah for Fajr prayer. This depends on the gathering of people and their readiness for prayer, while being careful not to let the preferred time expire. The single Adhan for Fajr is the view of Ath-Thawri, Abu Hanifa, and Muhammad ibn al-Hasan. However, the majority of scholars believe in the permissibility of two Adhans for Fajr: one before the entry of the time to awaken people and prepare them, and the other after the entry of the time.