Dominion: Cornucopia

No. Prizes are never part of the Supply and may not be bought or gained by cards such as Horn of Plenty; they may normally be gained only through Tournament or effects that gain cards from somewhere other than the Supply. Prizes have a cost of 0, which matters for cards such as Remake (p. 4).

Yes, any player who reveals a Province gains a Prize of their choice or a Duchy, but if another player also revealed a Province, the gained card goes on top of the deck instead of being drawn immediately. If you do not reveal a Province and another player does, you receive only +1 Action (p. 8).

When Young Witch is used, an additional Kingdom pile costing 2 or 3 is added to the Supply as the Bane pile. It can be bought or gained normally, and an empty Bane pile counts toward the game-end condition; each other player may reveal a Bane card from hand after Young Witch is played to avoid gaining a Curse (pp. 3, 9).

Horn of Plenty is a 0-cost Treasure and produces no coins. It gains a card costing up to 1 per differently named card currently in play, including itself, but if the gained card is a Victory card, Horn of Plenty is trashed (p. 6).

No. Hamlet lets you draw a card and get +1 Action, then you may discard one card for either +1 Action or +1 Buy, or discard two cards for both bonuses. You cannot discard extra cards to gain additional Actions or Buys (p. 6).

Menagerie gives +1 Action, then you draw only one card if any two or more cards in your hand have the same name; otherwise, you draw three cards. Only card names matter, so Copper and Silver are different cards (p. 7).