When Imp/Conclave are used to play a Duration action do they stay out with the duration (like Throne Room for example)?
Thanks
No. From an IRL point of view, they played the Duration card only once, so the Duration card itself is enough of a reminder (unlike Throne Room, where the Throne Room is to remind you that you get 2 effects next turn).
From a strict rules perspective, non-Duration cards stay out only under 2 conditions
1) they played a Duration card an additional time (Throne Room, etc.), or
2) they played a card from somewhere else than your hand and "leave it there", with some more restrictions that are irrelevant here (Necromancer, errataed Overlord, etc.).
Imp/Conclave do not fall into either category (excluding an edge case with Way of the Mouse), so they don't stay out.