...I don't understand what all of the controversy here is about.
1. A Duration card is an orange card that says Duration in on the bottom. If no Duration cards are involved, a card is discarded from play even if it is going to do something on the next turn (like Possession).
2. If a Duration card does have something to do after the end of this turn, it stays in play. This is true even even if it was played using Way of the Chameleon.
3. If a Duration card is played "leaving it there", never entering the play area, then the card that played the Duration card remains in play, even if it is not itself a Duration card, such as Band of Misfits, Estate, or any non-Duration card acting through Way of the Mouse.
The question at the top of this thread is about situation (2); the question is, does item (2) apply if the card is played according to Way of the Squirrel? For (2) to be relevant, it would have to be the case that drawing your next hand in Cleanup is considered "after the end of this turn" for Duration purposes. Jeebus is right that (in the context of Outpost) Donald did, a long time ago, answer that question in the affirmative. However, more recently (this past weekend) Donald said something that implicates the opposite: he said that the reason Outpost stays in play after a Lich turn is because trying (and failing) to give you an extra turn happens after the end of the current turn; he didn't say anything about the three-card hand being relevant to why or whether Outpost stays in play. The fact that he didn't mention it suggests (but doesn't prove) that it's not relevant, and drawing your next hand is no longer considered "after the end of this turn" for the purposes of Duration evaluation.