1) Being asleep is a special condition. And most Poke-Powers, if not all, say "This power can't be used if this pokemon is affected by a special condition". Meaning, you can't use Pokepowers while being asleep. PokeBodies, however, are always in effect, no matter what condition is on your Pokemon.
2) After putting your opponent to sleep, your opponent flips at the Beginning, and End of their turn (If their Pokemon is still asleep). If you put your opponent to sleep, you continue with your turn. When you end it, your opponent flips a coin. If it's heads, it is awake and able to attack. If tails, your opponents pokemon remains asleep for the rest of their turn. After your opponent ends their turn, they flip again. If heads, they are awake, if tails, they remain asleep for your turn, and this process repeats over and over again.
Now with Darkrai's attack "Dark Slumber", your opponent does his/her turn normally w/ whatever he/she decides to do. When they end their turn, your opponents pokemon is now asleep. They flip a coin and if heads, they are awake, tails still asleep.
Keep in mind however, I think if your opponent switches a to a new Pokemon, the affect of "Dark Slumber" is no longer in affect.