This is set right after he killed his wife/daughter, became the Ghost of Sparta and went from serving Ares to serving Athena. The chronological order for the games is (including this one but not counting a couple downloadable mini-games for PC/phone): GoW:A, GoW: CoO, GoW1, GoW: GoS, GoW2 and GoW3.
No. This game takes place after Kratos killed his family. Also the end of God of War 3 implies he survived his suicide attempt. Plus David Jaffe says in God of War 4 he's going to fight either the Egyptian or Norse gods.