My opponent attacks my Active, which causes 20 damage and Poisons it. Her turn is over. Do I then have to put 1 more damage [counter] on it because it is still Poisoned?
If you use an attack that puts your opponent's Pokémon to Sleep (like Haunter's Nightmare), and the Defending Pokémon was already Asleep before you attacked, do you still have to flip a coin after the attack to see if it wakes up?
Yes, they get to flip at the end of each player's turn.
If one of your Pokémon was Poisoned, and a few turns later it is KO'd by the Poison, does your opponent take a Prize card although the Poison KO'd the Pokémon, and not your opponents Pokémon?
Yes, he would still draw a Prize card. Whenever a Pokémon is Knocked Out, the other player gets Prize cards -- even if you Knock Out your own.
When you Knock Out more than one of your opponent's Pokémon in a turn (i.e. using Selfdestruct or Raichu's Gigashock attack) do you receive a Prize for each Knocked Out enemy Pokémon?
Yes you do. And if you knock out some of your own, your opponent gets to draw prizes as well.
At the end of a long match, I Selfdestruct Magneton. My opponent and I both take our last Prize cards. Who wins?
Well, if one of you has a Benched Pokémon to replace your Active Pokémon and the other player doesn't, then the person who can replace his or her Active Pokémon wins. Otherwise, you play Sudden Death. This is explained in the Pokémon rules in the Expert Rules section under "What Happens if Both Players Win at the Same Time?"