Ooh, these are tough.
Mostly I can discern the outcomes via careful thought of the text.. but these simply seem ambiguous to me.
Possession text for reference: "The player to your left takes an extra turn after this one, in which you can see all cards he can and make all decisions for him. Any cards he would gain on that turn, you gain instead; any cards of his that are trashed are set aside and returned to his discard pile at end of turn."
Trader: Which side(s) can reveal trader upon the possessed player gaining a card?
This one is the tricky one... I think the answer is both. First, the Possessed player "would gain a card", and so they can reveal trader. The "would gain a card" wording is the same on Trader and Possession, I think; thus, the player whose turn it is (the Possessed player) can decide which order they happen in, and so can reveal trader. Obviously, the possessor can reveal their own trader after they gain the card instead of the Possessed player.
Noble Brigand: Who is the attack played on? If the possessed player plays the on-buy attack, does the player who played the possession gain the cards back (which would be oddly hilarious).
Yep, I think that is what happens. On-buy, there's an attack - it affects all other players (including the possessor). However, if the Possessed player would gain any cards with that attack, then those cards go to the possessor instead. So yep, the hilarious option is what happens.
Ill-Gotten-Gains: Who gains the curse? If the possessed player gains the curse, which I think is the case (on-gain wording matches possession, unlike NB's on-buy) does the cursed get sent to the player who played the possession?
Both the IGG itself and the curse go to the player who played the Possession. All cards gained goes to the player who played the Possession, including stuff they don't want like curses.