Short answers, pulled from the story.
Mysterium Cosmographicum is an astronomy book by the German astronomer Johannes Kepler, published at Tübingen in late 1596. Its title translates variously as The Cosmographic Mystery, Cosmic Mystery, or The Secret of the World. A second, expanded edition appeared in 1621.
Kepler proposed that the distance relationships between the six known planets could be explained by nesting the five Platonic solids inside one another, each encased in a sphere. Ordered as octahedron, icosahedron, dodecahedron, tetrahedron, and cube, the resulting spherical shells corresponded to the relative sizes of planetary orbits, varying from observed measurements by less than ten percent.
Kepler described having an epiphany on the 19th of July, 1595, while teaching in Graz. He was demonstrating the periodic conjunction of Saturn and Jupiter in the zodiac when he realized that regular polygons bind inscribed and circumscribed circles at definite ratios, a relationship he thought might underlie the geometry of the cosmos.
Tycho Brahe said the ideas were intriguing but could only be verified through the observational data he had been collecting over the previous thirty years. He had promised Kepler access to those observations, which led Kepler to seek him out at the beginning of 1600. Brahe shared only the data on Mars, but that exchange helped Kepler formulate his laws of planetary motion.
Theology was central to the book's argument. Kepler believed he had revealed God's geometrical plan for the universe. He understood the cosmos as an image of the Trinity, with the Sun corresponding to the Father, the stellar sphere to the Son, and the intervening space to the Holy Spirit. His original manuscript included a chapter reconciling heliocentrism with biblical passages, which the Tübingen university senate required him to remove before publication.
Yes. In 1621, twenty-five years after the first printing, Kepler published an expanded second edition that was half as long again as the original. It included extensive footnotes documenting the corrections and improvements he had made in the intervening decades, though he never abandoned the Platonic solid framework at the book's core.