The reply from drsteelhammer is wrong, because it doesn't take into account the fact that a Possession turn is always played by the Possessed player. That is, it's the Possessed player's turn, not the Possessor's turn. EDIT: cross-post with Stef.
What happens if...
I play someone's Outpost while Possessing them once?
So you mean that
they play Outpost (as chosen by you). They draw 3 cards in Clean-up, then get an extra turn from Outpost (since they've only had one turn before, the extra turn from Possession). They control this turn. Then they get their normal turn.
I've Possessed them multiple times and play their Outpost during the first Possession turn?
So you mean that
they play Outpost. They draw 3 cards in Clean-up. Then after this turn, they (not you*) decide whether to take the Outpost turn or a Possession turn. If they choose Possession, the 3 card turn will be controlled by you, then the Outpost turn won't happen, since they already had two turns. If they choose Outpost, the 3 card turn will be controlled by them. Then the rest of the Possession turns happen.
*Donald recently changed this ruling. Before the Possessor got to choose in between turns....the last Possession turn?
...a/the middle Possession turn?
In either case, they draw 3 cards in Clean-up, but don't get an extra turn, since they already had two or more turns. EDIT: As clarified by Stef, if it's the
last Possession turn, their regular turn will be the 3 card turn. If not, it will be one of the Possession turns controlled by you.
...two Possession turns in a row?
The second time, they draw 3 cards in Clean-up, but don't get an extra turn, since they already had two or more turns.
I play Possession and my own Outpost in the same turn?
Your extra turn (from Outpost) always comes first, since you're the active player and effects go in turn order.
I'm Possessing someone and I play their Possession and their Outpost?
So you mean that
they play Possession and Outpost. Their extra turn (from Outpost) always comes first, then the
next player's extra turn (from Possession). The next player is the player after them; it's you if it's a two player game. After that turn, they (the player you Possessed) get their normal turn.
The above happens on the first Possession turn of a multi-Possession?
They draw 3 cards in Clean-up. Then after this turn, they decide whether to take the Outpost turn or a Possession turn. If they choose Possession, the Outpost turn won't happen, since they already had two turns. Then the rest of their Possession turns happen. After that the
next player's Possession turn happens. After that turn, they (the player you Possessed) get their normal turn.
...the last Possession turn?
...a/the middle Possession turn?
In either case, they draw 3 cards in Clean-up, but don't get an extra turn, since they already had two or more turns. After any and all of their Possession turns, the
next player's Possession turn happens. After that turn, they (the player you Possessed) get their normal turn.
...two Possession turns in a row?
The second time, they draw 3 cards in Clean-up, but don't get an extra turn, since they already had two or more turns. After any and all of their Possession turns, the
next player's two Possession turns happen. After that turn, they (the player you Possessed) get their normal turn.
And finally, can I replace "play Outpost" with "buy Mission" and get similar answers (as in, replace starting with a 3-card hand with not being able to buy cards) or would there be a difference?
The difference is if you Possessed them multiple times and they buy Mission:
If it's the first Possession turn, they will get the Mission turn, since their previous turn wasn't theirs. After this turn, they decide whether to take the Mission turn or a Possession turn. In any case the Mission turn will happen. If they choose Possession now, they get to make the same choice again afterwards (until the Mission turn is played or there are no more Possession turns).
If it's a later Possession turn, buying Mission will not get them an extra turn, since the previous turn was theirs.