I can add to the questions listed in the opening posting:

Q: Do I need a Gold Rating, or will Silver and Bronze Ratings give me the Atma as well?

I got an Atma with Bronze, landing merely 3 hits just before the FATE was over so participation is enough just like with the beastmen FATES

Q: I have multiple classes with Zenith Weapons, will I have to be on the job that started the original quest, or can I choose which one to upgrade? i.e. If I started the quest on White Mage, but want to upgrade my Bard Zenith Weapon instead, can I do it?

I started the quest as BLM but got annoyed from doing Low Level FATES with him so I switched to BRD and got 3 Atmas since this. So it doesn't matter at all, even on your first quest run. And I can only recommend BRD, he is KING of FATES.