Yes, he might. If Dr John is into Japanese pop culture or is a fan of modern literature, there's a good chance he has a light novel. Light novels are quite popular these days.
Since we don't know the exact story, it's difficult to name the key elements. However, common elements in a family - related story could be relationships, like love or conflict among family members. There might also be family traditions or secrets.
I haven't come across any information suggesting that 'Dr. John' has a corresponding light novel. Usually, such adaptations are announced or mentioned, but not in this case.
I'm not familiar with the story of Dr John A King. He could be a doctor in a particular field, perhaps medicine or academia. Maybe he made some significant contributions in his area of work, like conducting important research or providing excellent patient care. But without more details, it's hard to say exactly what his story is.
The 'family of the Dr John story' is quite a mystery without further details. Dr John could be a doctor in a medical drama, a character in a historical fiction, or something else entirely. His family might play important roles in the story, such as supporting him in his work, or causing some kind of conflict. But again, we really need more information to accurately describe them.