How does Emissary of Grudges work exactly?
Can I use Emissary of Grudges's ability as an instant?


Yes, otherwise it wouldn't make sense. A spell or ability is something on the stack.


  can I "cancel" Clone's ability, and use it myself on anyone?


Nope. Clone's ability doesn't target; the word 'target' has a very specific meaning in Magic and is almost always printed on the card, in the rules text or the reminder text. A similar card which does target is Rite of Replication, which does not target you but it might target a permanent you control. If it does, you can choose another target for that spell, but it must be a creature (for example, a Grizzly Bears; it doesn't matter who owns it) and the controller of the spell doesn't change. The cloned Bears will still be controlled by player 2.


  if Emissary of Grudges leaves the battlefield, do I keep this ability? Or do I just throw out the chosen player's name?


You don't need to reveal it, you can just throw the paper you wrote it on away. If it enters the battlefield again, you can choose a new player.