Is the intention (niyyah) obligatory when reciting the Qur’an, whether the recitation is outside of prayer or within it, and is the intention obligatory when reciting Al-Fatihah in prayer, specifically in His, the Almighty’s, saying: "Guide us to the straight path," with the intention of supplication?

Reciting the Quran does not require an intention because it is an act of worship that can only be performed as such, whether inside or outside of prayer. As for what is recited from the Quran during prayer, it does not require a specific intention for it; rather, the intention is for the prayer itself. There is no harm in intending to make supplication with a Quranic verse, as this is part of reflecting upon the Quran and interacting with it. The Prophet, peace and blessings be upon him, used to ponder the meanings of the Quran and would supplicate and seek refuge when passing by verses of mercy and punishment.