Psychic Possession's controller, not the enchanted player, skips their draw step.

An Aura with enchant opponent can't be attached to a permanent.

In a multiplayer game, if the enchanted opponent leaves the game, Psychic Possession will be put into its owner's graveyard as a state-based action. If Psychic Possession's owner leaves the game, Psychic Possession immediately leaves the game; this is not a state-based action.

If Psychic Possession's controller ever happens to be the player it's enchanting, Psychic Possession will be put into its owner's graveyard as a state-based action.

Enchanting an opponent works very much like enchanting a permanent. The Aura spell targets the opponent. When it resolves, it enters "attached" to that player. Once it's on the battlefield, it no longer targets that player.