When a Scepter plays a Duration card as a Way (such as Way of the Monkey), should the scepter stay in play the next turn? I saw it do this.
Yes, that's correct, assuming the original playing was for the normal effect of the Duration. The rules try to be simple and not consider to many "corner cases".
The rule is that if a card (like Throne Room or Scepter) plays a Duration card extra times, it stays out as long as the Duration card. In your case, the Scepter did play a Duration card an extrav time, so the Scepter stays out as long as the Duration card. Since a Duration card stays out generally for at least one turn when played "normally", so will the Scepter.