Short answers, pulled from the story.
The League of Nations was officially established on the 10th of January 1920. This date marks the entry into force of the Covenant of the League of Nations, which had been signed on the 28th of June 1919 as Part I of the Treaty of Versailles.
The headquarters of the League of Nations were established in Geneva, Switzerland. Geneva was chosen as a neutral country that had long been a hub for international humanitarian efforts, and the first meeting of the Council took place there on the 16th of January 1920.
The United States never joined the League of Nations despite being the primary architect of the organization. This decision severely weakened the authority of the League from the outset and contributed to its inability to prevent aggression by the Axis powers in the 1930s.
The League of Nations was designed with three main bodies: an Assembly representing all member states, a Council acting as an executive body with permanent and non-permanent members, and a Permanent Secretariat to handle administrative duties. The Council began with four permanent members and four non-permanent members elected by the Assembly for a three-year term.
The League of Nations formally dissolved on the 18th of April 1946. This dissolution occurred after the organization had been largely inactive since the onset of the Second World War in 1939, and many of its components were relocated into the new United Nations.