Does that line restrict it's use when TWF/MWF with weapons that each have it?
It appears yes. Each conductive can be used once per turn, but if you have multiple conductive weapons (either through fighting with 2 weapons or through the use of the Quick Draw feat between attacks) you can activate them each seperately. This is powerful, but expensive (both in monetary cost of adding the property to multiple weapons and the drain on abilities with limited uses per day).
And if you can TWF activating both weapons with the property, how does it work for AoMF? (given that Speed is the exception, not the rule)
When using an unarmed strike while having an amulet of mighty fists on you would gain the conductive of the item only once per turn, since your unarmed strike is a single weapon. If a creature had multiple natural weapons, each would be enhanced seperately and each natural weapon would be allowed to use the amulet in one turn, as is my understanding.