Short answers, pulled from the story.
The first session of the United Nations General Assembly convened on the 10th of January 1946 inside the Methodist Central Hall in London. Representatives from fifty-one founding nations gathered to establish a new global order after years of war.
The organization finally relocated to its permanent home in Manhattan on the 14th of October 1952 when the seventh regular annual session began. The assembly moved its operations to Flushing, New York, where it met at the former New York City Pavilion of the 1939 World's Fair until 1951.
Fifty-one nations joined the United Nations as founding members in 1945 before expanding to reach 193 current members today. Developing countries now constitute more than two-thirds of all member states and often determine the agenda through coordinating groups like the G77.
Budgetary concerns and elections for membership admission or expulsion also follow this higher threshold rule requiring a two-thirds majority of those present and voting. Resolutions passed by the assembly generally lack binding legal force over member states except regarding budgetary measures.
The Uniting for Peace resolution adopted on the 3rd of November 1950 allowed the body to take action if the Security Council failed due to veto power. This mechanism enabled immediate consideration of threats to international peace even when permanent members blocked direct council action.