Short answers, pulled from the story.
Halley's Comet is named after Edmond Halley because in his 1705 paper Astronomiae cometicae synopsis he identified that comet sightings in 1456, 1531, 1607, and 1682 were all the same object and predicted it would return in 1758. When the comet appeared on schedule, astronomers assigned his name to it. Halley himself died in 1742 and never witnessed the return he had predicted.
Yes. After Halley visited Isaac Newton in Cambridge in 1684 and recognised the importance of Newton's work on planetary motion, he arranged and personally funded the publication of the Philosophiæ Naturalis Principia Mathematica, which appeared in 1687. Halley was working as secretary of the Royal Society at the time, not in a position of significant personal wealth.
In 1717-18, Halley discovered the proper motion of the so-called fixed stars by comparing his own measurements with those in Ptolemy's Almagest. He found that Arcturus and Sirius had both shifted position; Sirius had moved 30 arc minutes southward over 1800 years, a distance equal to the apparent diameter of the Moon.
In 1692, Halley proposed that Earth consisted of a shell roughly 500 miles thick surrounding two inner concentric shells and an innermost core, each separated by its own atmosphere and rotating at a different speed with its own magnetic poles. He advanced this model to explain anomalous compass readings and speculated that gas escaping from the interior produced the aurora borealis.
Halley made three voyages aboard the Paramour under orders from King William III. The first, beginning in November 1698, was cut short by crew insubordination. The second, from September 1699 to the 6th of September 1700, covered the South Atlantic from 52 degrees north to 52 degrees south and produced the General Chart of the Variation of the Compass, published in 1701. A third voyage in 1701 studied tidal patterns in the English Channel.
Halley became the second Astronomer Royal of Britain in 1720, succeeding John Flamsteed. He held the position until his death in 1742 at the age of 85. He was buried at St Margaret's Church in Lee, in the same vault as the Astronomer Royal John Pond.