Is it obligatory to observe the proper sequence when making up missed prayers (Dhuhr and Asr), and should they be prayed shortened or in full upon arriving in one's country of residence?

You must make up for the Dhuhr and Asr prayers that you missed due to delaying them without a valid excuse, which is one of the major sins. You must make them up immediately and in their proper order (Dhuhr then Asr), performing them fully with four rak'ahs for each prayer. The missed prayers should be performed before the Maghrib prayer, unless doing so would cause the time for Maghrib to expire, as its time ends with the disappearance of the twilight redness.