Short answers, pulled from the story.
Since Charles Talbot, 1st Duke of Shrewsbury resigned in 1714, no individual has been appointed Lord High Treasurer. Instead, the office has been in commission, with a board of Lords Commissioners of the Treasury collectively carrying out its functions. The arrangement was first used as early as 1612 and became permanent after 1714.
The English Treasury came into existence around 1126 during the reign of Henry I, when financial responsibilities were separated from the broader duties of what became the Lord Great Chamberlain. A formal Treasurer was appointed in 1216 to take control of the Treasury at Winchester.
William Cecil, 1st Baron Burghley, served as Lord High Treasurer from 1572 to 1598. During his twenty-six year tenure he dominated the administration under Elizabeth I, effectively acting as a de facto prime minister. His career is the most prominent example of how powerful the office became in the 16th century.
The Lord High Treasurer received the office by the delivery of a white staff, while the companion office of Treasurer of the Exchequer was conferred by letters patent under the Great Seal. By the 18th century the installation involved a formal procession through Westminster Hall, oaths before the Lord Keeper of the Great Seal in the Court of Chancery, and a concluding ceremony at the Exchequer of Receipt where a tally was struck and the great keys of the Treasury were presented.
The two offices were merged by the Consolidated Fund Act 1816, which united them into a single title on the 5th of January 1817. The United Kingdom of Great Britain and Ireland had been created in 1801, but the consolidation of the treasury offices did not follow until that Act took effect.
The Lords Commissioners of the Treasury collectively execute the office. By modern convention the prime minister serves as First Lord of the Treasury, the Chancellor of the Exchequer as Second Lord, and government whips in the House of Commons are appointed as Junior Lords Commissioners. These are sinecure appointments tied to their other governmental roles.