Is it obligatory for one who is stoning the Jamarat on their own behalf and on behalf of another to start by stoning on their own behalf, then return to stone on behalf of the one who appointed them, or should they stone on their own behalf then on behalf of the one who appointed them at each Jamarah?

The proxy must begin by stoning on his own behalf, then stone on behalf of the one who deputized him. What is favored is that he stones the first pillar (jamrah) on his own behalf, then stones it on behalf of the one who deputized him, and so on. Among the reasons for this preference is that if one stones on behalf of another but does not stone on his own behalf, the stoning counts for the one who deputized him, not for the proxy himself. This is easier due to the severe crowding. Allah Almighty says: "And He has not placed upon you in religion any difficulty" (Quran 22:78).