Is it valid to hasten in throwing the pebbles on the first day of departure (Nafar al-Awwal) with the intention of leaving Mina before throwing the Jamrat al-Aqaba al-Kubra, then throwing it and not returning to Mina; and is it obligatory to return to Mina after throwing and then intending to depart?

There is no issue upon the questioner due to the non-obligatory nature of returning to Mina. As for the Shafi'is' stipulation that the intention must accompany the departure, it is fulfilled for everyone, and the intention is sufficient when commencing the departure before Jamrat al-'Aqaba, which is what the questioner did. The Shafi'i jurists have explicitly stated that the intention accompanying the undertaking of departure preparations is sufficient, even if one forgets it after completing them and before reaching the Jamra. It is normally impossible for a sane, free-willed person to engage in packing without conceiving of what he is packing, and to head towards the road to Mecca without noticing and intending to reach it.