Short answers, pulled from the story.
The Bessemer process is named after the English inventor Henry Bessemer, who patented the method in 1856. American inventor William Kelly claimed to have discovered a similar process independently around 1851, though historians regard the claim as controversial and consider Kelly's version less developed than Bessemer's.
The Bessemer process allowed mass production of steel from molten pig iron at a fraction of the previous cost, dropping the price from forty pounds per long ton to six or seven pounds. It reduced the time to convert iron to steel from at least a full day to ten to twenty minutes, making steel economical for railroads, bridges, and large-scale construction.
English metallurgist Robert Forester Mushet solved the quality problem after conducting thousands of experiments in the Forest of Dean. His method involved burning off all carbon and impurities, then reintroducing carbon and manganese by adding a precise amount of spiegeleisen. Mushet's patent later lapsed and was acquired by Bessemer.
The basic Bessemer process, also called the Gilchrist-Thomas process, replaced the clay converter linings of the original acid process with dolomite or limestone linings. This change allowed removal of phosphorus from high-phosphorus iron ores, which the acid process could not achieve. A patent for the method was issued in May 1878 to Sidney Gilchrist Thomas and Percy Gilchrist.
After visiting Bessemer's works in 1872, Andrew Carnegie commissioned Alexander Lyman Holley to build the Edgar Thomson Steel Works, which opened in 1875. Carnegie Steel used the process to cut the price of steel railroad rails from one hundred dollars per ton in 1873 to eighteen dollars per ton by the 1890s.
In the United Kingdom, the process was described as declining as early as 1895. In the United States, commercial steel production using the Bessemer method stopped in 1968. It was replaced primarily by the basic oxygen Linz-Donawitz process, which offered better control of steel chemistry and handled scrap steel more efficiently.