Note: The owner of gpuzzles.com did not allow me to copy the puzzle, However site is fine pasting picture.
All done according to logic stated in question. Actual legal aspects are a different story (I'm thinking manslaughter would be closer than murder)
Not suicide, since his actions didn't cause his death.
Intended to kill someone, and by his action, killed someone else. Thus, guilty of murder. Note: Didn't pick it up the first time, but the story mentions "only threatening", implying he "knew" the gun was empty. However, he did pull the trigger, which contradicts the threatening.
Caused death of Alexis by actions intended to kill someone, thus, guilty of murder.