Is it permissible to pray immediately after the Azan without waiting, knowing that I do not know the Muezzin but I believe he calls the Azan at the correct time?

There is no harm in entering into prayer immediately after hearing the Adhan from a trustworthy muezzin, because he only calls the Adhan after the time for prayer has begun, as the Prophet, peace and blessings be upon him, said: "The muezzin is entrusted."