Is it permissible for a Muslim to ask others to supplicate for him (make du'a)? And have any weak hadiths been narrated concerning this, or is it an innovation, or is it a legitimate matter?

It is commendable for a Muslim to ask his Muslim brother for righteous prayers, especially if the brother is among the righteous. Among what has been narrated regarding this is the saying of the Prophet, peace and blessings be upon him: "The supplication of a Muslim for his brother in his absence is answered. At his head there is an appointed angel; whenever he supplicates for his brother, the appointed angel says: Amen, and for you the like thereof." And his saying to Umar: "Do not forget us, my brother, in your supplications."