QIf I have an amulet that costs 3 play points or less in my hand, do I have to put it onto my field with this follower's Fanfare ability?

ANo. The hand is a non-public zone, so there is no way to verify that you have a card that meets the condition, and therefore you can choose to not put the amulet onto your field.

QIf this follower is on my field, and 2 amulets come onto my field at the same time, does this follower's "Whenever an amulet is put onto your field" ability trigger twice?

AYes. It triggers once for each amulet that comes onto your field.