Not at all. Yuji Itadori's story doesn't end with his execution in the manga. There are many developments and challenges he faces, but execution isn't one of them.
Yuji Itadori survives till the end. The manga doesn't have his death as the final outcome. The story might have thrown challenges at him, but he manages to overcome them and remain alive.
Definitely. In the Hitman 2 Expansion New York, the story is an important aspect. It sets the context for the various assassination contracts. For example, there might be a power struggle within a certain organization in New York, and the targets are key players in this. The story also provides a backdrop for the unique locations and interactions available in the New York setting, making the gameplay more immersive.
Itadori did indeed die in the manga. His death had a significant impact on the story and the other characters. It was a turning point that led to many new developments in the plot.