Short answers, pulled from the story.
MI5 was founded in October 1909 as the Home Section of the Secret Service Bureau, a joint initiative of the Admiralty and the War Office. It became Military Intelligence section 5 in September 1916, though it held that name officially for only thirteen years before being renamed the Security Service in 1931.
Vernon Kell of the South Staffordshire Regiment was the founding head of MI5, serving from 1909 until the early part of the Second World War, when Winston Churchill sacked him shortly after coming to power in 1940.
The double-cross system was a wartime deception operation in which MI5 turned captured German agents into double agents, using them to transmit false intelligence back to the Abwehr. It began with agent Arthur Owens, codenamed Snow, and was coordinated by the Twenty Committee. A post-war analysis found that of roughly 115 agents targeted against Britain, all but one were caught.
The Cambridge Five were Kim Philby, Donald Maclean, Guy Burgess, Anthony Blunt, and John Cairncross, recruited by Soviet intelligence from Cambridge University. All except Maclean served with MI5 or MI6 during the Second World War, and the ring passed more than 16,000 documents to the Soviets before being exposed after the war.
MI5 is headquartered at Thames House on Millbank in London, where it has been based since 1994. The building also houses the Joint Terrorism Analysis Centre (JTAC). The service's current postal address is PO Box 3255, London SW1P 1AE.
In March 2018, the UK government acknowledged that MI5 officers are permitted to authorise agents to commit criminal activity within the United Kingdom. The Investigatory Powers Tribunal dismissed a legal challenge to this policy in December 2019 in a 3-to-2 decision. A Bloomberg report indicated the authorised activities can include murder, kidnap, and torture.