Kairi die in Kingdom Hearts

Answer 1

No, she goes back to the island after being separated by Riku then Sora.

True, but she had lost her heart through the whole game. Sora almost dies, however, because he gave his heart (which contained Kairi's within) so she could live and be safe--his whole purpose on the adventure. He does get it back when Kairi shields him from the Heartless attack (even though he was a Heartless as well, but Kairi suspected it was him from the start). So, Kairi does not die...she goes back home, to answer your question.