Is it permissible for the worshipper to say "A'udhu billahi min ash-shaytanir-rajeem" (I seek refuge in Allah from the accursed Satan) before the Imam begins Al-Fatiha, then listen to his recitation, and only after that say "Bismillah" (In the name of Allah), knowing that the Imam is quick in his prayer? And is it permissible to omit "Bismillah" in a recitation where you do not consider it part of Al-Fatiha?

The 'isti'adha' (seeking refuge with Allah) before Al-Fatiha is recommended by the majority of scholars, and whoever omits it, their prayer is valid. The general rule is that the 'isti'adha' should be recited at the beginning of the recitation, as stated by Al-Qurtubi and Ibn Ashour. There is no evidence to prohibit the 'isti'adha' in the manner mentioned in the question. Reciting Al-Fatiha after listening to the Imam is more preferable to avoid the disagreement of scholars who deem it obligatory. It is permissible to omit the 'basmala' if one recites according to a narration that does not consider it part of Al-Fatiha, as many scholars believe it is not a verse from it.