By the end of both the anime and the manga Inuyasha, Kagome doesn't die.
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in episode 13 when he thouht he was going to die he said that she smelled good and wanted to lay on her lap but later on he does confess they kiss in movie 2
Kagome doesn't kiss anyone but Inuyasha. As for Inuyasha he has kissed both Kikyo and Kagome.
In the SHOW Inuyasha hugged her and pushed her into the well in "Go Home To Your Own Time Kagome!" So CUTE! Reccomend me, Kagome Higurashi, as a answerer!
when Kaoru Wada wrote it, I think he just called it 'Kagome and InuYasha's Theme'.
well he does saves kagome and rin in movie 3