QIf I evolve an Eidolon of Madness on my field, does this follower's "when an amulet is put onto your field" ability trigger?

ANo, it doesn't.

QIf an amulet is put onto my field during my opponent's turn, does this follower's "once per turn, when an amulet is put onto your field" ability still trigger?

AYes, it triggers regardless of whose turn it is.

QI have 2 copies of this follower on my field. If an amulet is put onto my field, do the "when an amulet is put onto your field" abilities of both copies trigger?

AYes, they do.