Your only way to win is to be lynched. If you do, the player who nominated you will die as well.
Question and answers
If the Hunter is killed by the Jester, does she then still kill?
Yes. If she dies, she attacks.
Does a Jester win if a Poppet gets him lynched?
Absolutely. To him, a lynch is a victory, no matter how it happened.
Can the Jester choose to spare his nominator and not kill him?
No, the killing triggers automatically.