Can I play/use a card that can't fully resolve?
Yes! From the rulebook, page 7:


  When resolving a card ability, resolve as much of the ability as can
  be resolved, and ignore the rest.


In the Anger example, you would first gain the aember from playing the card. Then, if you have one or more friendly creature in play, you must choose one. If that creature is exhausted, ready it. If there is one or more valid fight targets, your chosen creature fights one of them. But any one of these bits fizzling doesn't prevent the others from happening. If there are no valid targets, you still get to ready a friendly creature. If you have no exhausted creatures, you can still have one of your ready creatures fight as a result of Anger. If you have a stunned card, Anger can remove its stun, as the stun removal replaces the fight use, even if your opponent has no creatures on the board.

FFG further clarified this in this video.