If a nontoken creature dies at the same time as Ulvenwald Mysteries leaves the battlefield, the first ability triggers. Similarly, if Ulvenwald Mysteries isn’t a token and has become a creature, it dying will cause its own first ability to trigger.

The last ability is a triggered ability, not an activated ability. It doesn’t allow you to sacrifice a Clue whenever you want; rather, you need some other way of sacrificing it, such as the activated ability that Clue tokens have.

If you sacrifice a Clue as part of casting a spell or activating an ability, the last ability will resolve before that spell or ability.