Must one repeat the prayer if one doubts having omitted the word "ʿalayhim" (upon them) at the end of Al-Fatiha in the first rakʿah, and is this considered a recurring doubt (shakk mustankah) that necessitates performing an alternative rakʿah?

If doubt occurs in every prayer or most of them (an obsessive doubt), it should be disregarded, whether it concerns forgetting a verse or any other pillars and obligations. There is no need to repeat the prayer or add another bowing (rak'ah). This does not change the ruling on frequent doubts, as scholars have explicitly stated that they should not be considered and should be ignored. Therefore, whoever habitually experiences doubt in their prayer must not pay attention to these doubts and whisperings, and should seek refuge with Allah from the accursed Satan, because they are from the plots of Satan. Similarly, if the doubt is merely an illusion with no basis, or if the doubts are numerous and repetitive, or if the doubt occurs after the completion of the act of worship (unless certainty about the matter is attained), it should be disregarded. It is advised to ignore all whisperings, especially those related to recitation, and to focus on contemplating the meanings of the Quran and experiencing humility (khushu') in prayer.