The story in the lyrics often depends on the theme and mood the songwriter wanted to convey. Sometimes it's a personal experience, other times it could be fictional or inspired by a broader concept.
Well, the extent to which Chainsaw Man follows the manga can vary. It might go quite far if there's enough demand and the story continues to engage readers and viewers. But sometimes, changes in the adaptation process can also influence how much of the manga is covered.
Invincible delves pretty far into the comics. It follows the main storylines and character developments quite closely. However, there are some differences and additions to make it work better in the medium of animation.
It depends on what you mean by 'far'. The Akatsuki manga has a considerable number of chapters and covers a wide range of storylines and character developments.