QIf my opponent and I each have a copy of this amulet on our fields, at the start of my main phase, can I choose to resolve my opponent's copy's ability before my own?

ANo. The active player always resolves their abilities before the non-active player. In addition, if a follower with Fanfare is put onto your field with this amulet's ability, you will need to resolve that Fanfare ability before your opponent resolves their Phoenix Roost's ability.

QIf there is more than 1 copy of this amulet on our fields, does each copy's ability get played?

AYes. All of them are played.