Is it obligatory to make up missed Eid prayers? And are the extra takbeers (utterances of "Allahu Akbar") a condition for them? And does the obligation to make up the prayers cease if the proper order is not observed between the missed prayers when making them up?

Scholars are of the view that making up a missed temporary supererogatory prayer (nafl) is sunnah (definition), and it is legislated to make up the Eid prayer for whoever missed it. It is preferable to make it up in its original form, including the extra takbirat, whether performed individually or in congregation. As for the order of making up missed prayers, the Shafi'i madhhab (definition) does not require ordering them, and this is more lenient for those who have many missed prayers.