Short answers, pulled from the story.
Flight controllers monitor specific technical systems of a spacecraft in real-time using telemetry, work from mission control centers such as NASA's Christopher C. Kraft Jr. Mission Control Center, and advise the flight director on decisions within their area of expertise. Each controller holds a unique call sign tied to their console, not their personal identity, so the position continues across shift changes.
CAPCOM stands for capsule communicator, a title dating to Project Mercury when the spacecraft was called a capsule. NASA designated CAPCOM as the sole voice communicating directly with the crew during crewed spaceflights, and filled the role with astronauts because NASA believed an astronaut would best understand conditions aboard the spacecraft and pass information most clearly.
During the Apollo 11 Lunar Module descent, "1202" and "1201" program alarms indicated a computer overload. Guidance officer Steve Bales consulted the guidance backroom, especially Jack Garman, who determined the problem was intermittent and could be ignored. Bales called "Go!", flight director Gene Kranz accepted the call, and the mission proceeded to a successful landing.
Jenny Howard Stein was the booster officer during Space Shuttle mission STS-51-F. When a second main engine showed indications of failure during ascent, which would have caused a mission abort, she determined the readings were a sensor error rather than an actual engine problem. At her direction the crew inhibited the sensor, saving the mission and possibly the crew.
The backroom, now officially called the multi-purpose support room or MPSR, is a set of support rooms staffed by flight controllers who are experts in specific system details and feed recommendations to the frontroom controllers at the main consoles. Information flows from the backroom to the frontroom to the flight director, and then potentially to the crew.
No. The flight operations directorate, or FOD, represents senior management at Johnson Space Center and advises the flight director on decisions with cost or public perception consequences, but the FOD cannot overrule the flight director during a mission. The FOD position was renamed from mission operations directorate when the flight crew operations directorate was merged back with it beginning in August 2014.