Short answers, pulled from the story.
The square is named after the British naval victory over France and Spain on the 21st of October 1805 off the coast of Cape Trafalgar in southwest Spain. The name was chosen around 1835, as suggested by architect George Ledwell Taylor, to commemorate Vice-Admiral Horatio Nelson's victory during the Napoleonic Wars. The square had earlier been proposed to bear the name of King William IV.
Trafalgar Square opened to the public on the 1st of May 1844. Planning began as early as 1826 when John Nash was instructed to draw up clearance plans, but Nash's death in 1835 stalled progress. Charles Barry's revised design was accepted in April 1840 and construction started within weeks.
The city of Oslo has donated a Christmas tree to Trafalgar Square every year since 1947 as a gesture of gratitude for British support during the Second World War. Prince Olav and the Norwegian government lived in exile in London throughout the war. The tree, selected by the Head Forester from Oslo's municipal forest, is shipped to the Port of Felixstowe and transported by road to the square.
Nelson's Column stands at 145 feet 3 inches. The original winning design by architect William Railton proposed a height of 218 feet 3 inches, but the column was built shorter following widespread public objections. Construction began in 1840 and the statue of Horatio Nelson was raised in November 1843.
The fourth plinth is the north-west plinth in Trafalgar Square that stood empty from the 19th century until 1999, when the Royal Society of Arts initiated a scheme to display rotating temporary artworks there. The programme is continued by the Fourth Plinth Commission, appointed by the Mayor of London. The other three plinths hold permanent statues, including equestrian figures of George IV and Charles I.
Trafalgar Square has hosted demonstrations including the Chartist rally of 1848, the Bloody Sunday riot of 1887, the Campaign for Nuclear Disarmament's first Aldermaston March in 1958, a 10,000-strong Vietnam War protest in March 1968, and the 200,000-person demonstration that sparked the Poll Tax Riots in 1990. Suffragettes bombed the square in 1913 and 1914 as part of their campaign for women's suffrage.