These etiquettes were put in place as a constant reminder to Muslims that those who are bestowed with the favours of Paradise will receive their account in their right hand on the Day of Judgement. This is the very reason eating with right hand was initiated by the Prophet (sws) as Sunnah and in addition to that, He always preferred his right hand over left in other matters as well. Regarding specific sequence of right and left on certain occasions, its seems that He preferred right to begin something and also preferred to leave it last. But principally, preference of right hand while eating has been proved as Sunnah, and same preference of right can be extended to other matters as well.

Please also refer to the video below from 40:54 to 43:06