Is it valid for the mosque's muezzin to say "Bismillahi Ar-Rahmani Ar-Rahim" (In the name of Allah, the Most Gracious, the Most Merciful) in a low voice into the microphone before the Adhan (call to prayer) and before the Iqamah (second call to prayer)?

There is no evidence in the Sunnah indicating the permissibility of commencing the Adhan with the Basmala. Therefore, one should not consistently do so, because the best guidance is the guidance of Muhammad, peace and blessings be upon him, especially since some ignorant people might think that this is one of the Sunnahs or obligations of the Adhan. It has been stated in the fatwas of the Permanent Committee that there is no known basis indicating the permissibility of seeking refuge (isti'adha) and uttering the Basmala before the Adhan.