Should a traveler who prayed Dhuhr seated in a bus facing away from the Qibla, despite being able to ask the driver to stop, repeat the Dhuhr prayer by shortening it? And would this entail repeating the Asr prayer which he prayed in the mosque before repeating Dhuhr?

First, the obligatory prayer is not valid while sitting if one is able to stand, as standing is a pillar. A worshiper on a bus can pray standing in the aisle.

Second, facing the Qibla in obligatory prayers is a condition for those who are able. If it is not possible to face the Qibla on the bus, the prayer is not valid unless there is a fear of the time expiring, and it is not possible to combine the prayer with another, and the driver refuses to stop. You did well by inviting your friend to repeat the prayer.

Third, whoever prayed Asr and then repeated Dhuhr due to its invalidity is not required to repeat Asr, because the Asr prayer was valid based on his belief at that time that Dhuhr was not obligatory upon him. He is considered as one who forgot, and the order is waived for him. Therefore, his prayer is valid, and he is not required to repeat Asr.