Did Jacob Grimm ever author a book on fairy tales?
No, he edited Grimms' Fairy Tales
Yes, he wrote Grimms' Fairy Tales
No, he never worked with fairy tales
Yes, he wrote Deutsche Mythologie
Answer
The answer to the question "Did Jacob Grimm ever author a book on fairy tales?" is no. Although the brothers Grimm are often credited with writing the fairy tales, they actually collected and edited them. They did not write any of the stories themselves.
Unraveling the Enchanting World of Jacob Grimm: A Mythical Quest in English