Short answers, pulled from the story.
Samuel Pepys, then president of the Royal Society, authorized the Principia with his formal imprimatur on the 5th of July 1686. Edmond Halley paid for the publication himself, at his own financial risk, after the Royal Society had exhausted its book budget on another work. The book appeared in summer 1687.
The Principia opens with "Definitions" and "Axioms or Laws of Motion" before proceeding into its three books. These axioms establish the laws of motion that form the mathematical foundation of classical mechanics; the precise names and numbering familiar today derive from Newton's formulations in that opening section.
Hypotheses non fingo means "I frame no hypotheses." Newton wrote the phrase in the General Scholium, added to the second edition of 1713, in response to critics who accused him of introducing occult forces into science by positing gravity acting across empty space without a visible medium. Newton's position was that the phenomena implied the attraction and that it was improper to speculate about causes not themselves implied by the phenomena.
Newton published three editions: the first in 1687, a second in 1713 with corrections, and a third on the 25th of March 1726. The first edition was printed in approximately 500 copies, the second in 750 copies, and the third in 1,250 copies.
In 1686, when Book 1 of the Principia was presented to the Royal Society, Hooke claimed Newton had taken the idea of an inverse-square law of gravity from him. Newton denied Hooke credit, arguing that Hooke had no mathematical demonstration and no evidence, and that prior figures had also advanced similar ideas. Halley reported that Hooke himself agreed the mathematical demonstrations of the resulting curves were wholly Newton's.
Émilie du Châtelet made the only complete French translation of all three books and their prefaces. Her translation added a commentary section synthesizing the three books and an analytical section applying calculus to Newton's most controversial theories. It remains the standard French translation.