Short answers, pulled from the story.
Apollo 5 was launched on the 22nd of January 1968 from Launch Complex 37B at Cape Kennedy. It was an uncrewed test flight of the Apollo Lunar Module, designated LM-1, designed to verify the module's systems before astronauts flew it.
The Apollo Guidance Computer aborted the burn because the spacecraft was not accelerating as quickly as the software expected. A valve had not been armed until ignition to prevent a suspected leak, causing propellant to reach the engine more slowly than programmed. The propellant tanks were also only half full, compounding the delay.
Gene Kranz served as flight director for Apollo 5. When the planned automated burn failed, Kranz and his Mission Control team in Houston devised an alternate plan and completed all engine tests under manual control.
The 'fire in the hole' test verified that the lunar module's ascent engine could fire while still physically attached to the descent stage. It simulated the procedure for an aborted lunar landing, requiring the descent stage to be shut down and control transferred to the ascent stage before ignition. The name came from a phrase used in mining when explosives are about to be detonated.
Grumman, based in Bethpage, New York, built the Apollo Lunar Module. NASA awarded Grumman the contract on the 7th of November 1962 after inviting eleven companies to bid.
LM-1's ascent stage re-entered the atmosphere on the 24th of January 1968 and burned up. The descent stage remained in orbit longer, re-entering on the 12th of February and falling into the Pacific Ocean several hundred miles southwest of Guam.
Yes. Apollo 5 was successful enough that a planned second uncrewed lunar module test using LM-2 was cancelled. This advanced NASA's timeline toward landing astronauts on the Moon before the end of the 1960s. The first crewed lunar module flight took place on Apollo 9 in March 1969.