Short answers, pulled from the story.
Free France was established as a government-in-exile in London in June 1940, following the Fall of France to Nazi Germany. It was led by General Charles de Gaulle, who had been promoted to brigadier general on the 1st of June 1940 and appointed Under Secretary of State for Defence on the 5th of June.
The Appeal of the 18th of June was a BBC radio broadcast in which de Gaulle urged French soldiers, sailors, and airmen to continue fighting. He declared that France was not alone and could, together with the British Empire, form a bloc controlling the seas and draw on American industrial resources. The broadcast was not widely heard in France that day but came to be remembered throughout France and its colonial empire as the voice of national honour.
By May 1945, Free French forces comprised 1,300,000 personnel, including roughly forty divisions. This made France the fourth-largest Allied army in Europe, behind the Soviet Union, the United States, and Britain. Approximately 58,000 men were killed fighting in the Free French forces between 1940 and 1945.
The Battle of Bir Hakeim took place from the 26th of May to the 11th of June 1942 in the Libyan desert, where General Marie Pierre Koenig and the 1st Free French Infantry Brigade held their position against the Afrika Korps under Generaloberst Erwin Rommel. Though eventually forced to withdraw, the defence was so formidable that British General Claude Auchinleck praised it publicly and even Adolf Hitler reportedly told a journalist returning from the battle that the French were the best soldiers after the Germans.
The Cross of Lorraine was adopted as the symbol of Free France at the suggestion of Capitaine de corvette Thierry d'Argenlieu. It was chosen to recall Joan of Arc, patron saint of France, whose symbol it had been, and to reference Lorraine, the province of her birth that had been partially annexed by Nazi Germany. Vice Admiral Émile Muselier incorporated it into the Free French naval jack on the 3rd of July 1940, two days after assuming command of naval and air forces.
Paris was liberated on the 25th of August 1944, when the military governor Dietrich von Choltitz surrendered to Allied forces led by General Leclerc's 2e Division Blindée. The division comprised roughly 10,500 French, 3,600 Maghrebis, and about 350 Spanish Republican exiles. Captain Raymond Dronne commanded the advance party given the honour of entering Paris first. The 1er Bataillon de Fusiliers Marins Commandos, who had landed on Sword Beach, were also among the first Free French units to enter the city.