Is it permissible to recite the Qur’an according to the the order of its revelation, and is there a specific reference that clarifies the narrations from the Meccan and Medinan periods?

The current arrangement of the Quran was by divine command (tawqeef nabawi), and the Prophet (peace be upon him) and his Companions acted upon it in their recitation. As for reciting it according to the order of its revelation, that is not possible.