Short answers, pulled from the story.
Gesta Danorum was written by Saxo Grammaticus, a Zealander and provost of the church of Roskilde. The work is generally agreed to have been finished no earlier than 1208, the date mentioned in its preface.
Archbishop Absalon commissioned the work. Absalon's importance in the text is so great that Book 14 alone, which ends with his appointment as archbishop in 1178, makes up nearly one quarter of the entire sixteen-book work.
Books 3 and 4 of Gesta Danorum tell the story of Amleth, a Danish prince whose father is murdered by his uncle Fengi, and who feigns madness before avenging the killing. Shakespeare adapted this story as Hamlet, and the name Hamlet is an anagram of Amleth, though whether Shakespeare chose the anagram deliberately is unknown.
The first printed edition was published on the 15th of March 1514 in Paris by the printer Jodocus Badius. It was prepared by Danish translator Christiern Pedersen under the title Danorum Regum heroumque Historiae.
Only four fragments of the original manuscripts survive: the Angers Fragment, the Lassen Fragment, the Kall-Rasmussen Fragment, and the Plesner Fragment. All four are held in the Danish Royal Library in Copenhagen, and the Angers Fragment is the only one confirmed to be in Saxo's own handwriting.
Gesta Danorum is one of the oldest known written sources on the histories of Estonia and Latvia. Its final three books are also a major source for the history of the Polabian Slavs, the Pomeranians, and Slavic paganism, and Book 14 contains a unique description of the pagan temple on the island of Rügen.