Right, this was a great challenge and I managed to read the following - tho some books fit several prompts:
1 ~ with a character who heads a company.
A Darker Domain
- One of the main characters heads up a business empire. His daughter and grandson were kidnapped and disappeared, which is at the heart of the story.
Doctor Fischer of Geneva
- The less said about the plot, the better, but Doctor Fischer heads up companies that produce both candy and toothpaste. (Greene's sense of irony.)
2 ~ about an important person.
3 ~ with a character who was a real-life famous person.
Love Game: A History of Tennis
- Includes several stories / biographies about famous tennis personalities.
Agatha Christie: An Autobiography (no review, yet)
- This one I would also like to fit into prompt #2.
4 ~ about royalty or with royal characters.
- none -