Are the Ta'awwudh and Basmalah considered part of the beginning of supplication, and must the worshipper behind the imam (ma'mum) recite them silently in every unit of prayer (rak'ah)?

Neither the Isti‘adha (seeking refuge) nor the Basmala (reciting "Bismillah") are part of the du'a al-istiftah (opening supplication). The Basmala is recited silently in every rak'ah before the Fatiha, due to the scholarly difference of opinion as to whether it is an ayah of the Fatiha or not. As for the Isti‘adha, some scholars deem it preferable to recite it before the Fatiha and the Basmala in every rak'ah, based on the apparent meaning of Allah’s saying: "So when you recite the Qur'an, seek refuge with Allah" (Quran 16:98). This is the view of the Shafi‘i school and the choice of Ibn Hazm, Ibn Taymiyyah, and Al-Albani. Others, however, hold that it should be limited to the first rak'ah only. This is the prevailing view among the Hanbalis and was favored by Ash-Shawkani. The Prophet, peace and blessings be upon him, used to seek refuge in his prayer by saying: "A‘udhu billahi min ash-shaytanir-rajim, min hamzih wa nafkhih wa nafthih" (I seek refuge with Allah from Satan, the accursed, from his madness, his pride, and his evil poetry), and sometimes he would add: "A‘udhu billahi as-Sami‘ al-‘Alim min ash-shaytanir-rajim, min hamzih wa nafkhih wa nafthih" (I seek refuge with Allah, the All-Hearing, the All-Knowing, from Satan, the accursed, from his madness, his pride, and his evil poetry).