Is uttering "Aʿūdhu billāhi min ash-shayṭāni ar-rajīm" (I seek refuge in Allah from the accursed Satan) sufficient for a non-Muslim if they pronounce it, or is Islam and strong faith a prerequisite? And is it permissible for a worshiper to utter it during bowing, prostration, rising from bowing, and during recitation when experiencing misgivings (waswas)?

Acts of worship are only accepted from a Muslim. Nothing is accepted from a non-Muslim until he believes in Allah, as Allah the Almighty says: "And whoever desires a religion other than Islam, it will never be accepted from him, and in the Hereafter he will be among the losers." A disbeliever may be rewarded in this world for his good deeds, as Allah the Almighty clarified in the Quran. He is provided for in this world for the good deeds he did for Allah, but he will have nothing of them in the Hereafter. Allah may answer the supplication of a disbeliever, such as his seeking refuge with Allah from Satan, and that may be a cause for his guidance to Islam or his protection from the harm of Satan. As for a Muslim's seeking refuge, it is likely to be answered, and the answer is proportional to the sincerity and truthfulness. However, seeking refuge with heedlessness is likely not to be answered.