Any card you play with King's Court is played three times. This is always true. The difference is in how those effects are carried out. As with most Dominion questions, the answer is evident by just following the text on the cards carefully.
First consider Pillage. The first time you play it is easy. The second time, follow the text. It says "Trash this." Well, it's already trashed. KC is just playing a phantom copy or something, so you can't trash it again. Then it says "Each other player..." and so on. You carry out all those effects, and you do it again for the third play.
Now consider Madman. The first time you play it is easy. The second time, it says "+2 actions". OK, you get +2 actions. Then it says, "Return this to the Madman pile." Well, you already did and you cannot do it again. So let's read on. "If you do, +1 Card per card in your hand." Oh, but you didn't! Since you did not meet the condition, you do not get that second part. So the result of KC-Madman is +6 actions and double your original hand size.
Knights take effect all 3 times, even if they get trashed in one of the plays.
Peddler says that it costs less per Action card you have in play, so if a card gets trashed or returned to the supply as a result of getting played, it will no longer count towards Peddler's cost reduction. KC-Woodcutter would only reduce Peddler down to $4 because only 2 cards end up in play, even though Woodcutter gets played thrice.
Conspirator only cares about played action cards, not what remains in play, so Conspirators will be activated after KC-Madman even though the Madman disappears.