Short answers, pulled from the story.
The Wehrmacht armies invaded Belgium, Luxembourg, the Netherlands and parts of France on the 10th of May 1940. This campaign marked the end of a period known as the Phoney War that had begun after Germany's invasion of Poland on the 1st of September 1939.
Maurice Gamelin served as the French commander-in-chief during the Battle of France. He believed the Ardennes region never favoured large operations despite General Philippe Pétain declaring it impenetrable for any large-scale invasion force.
On the 22nd of June 1940 the Second Armistice at Compiègne was signed by France and Germany. The fascist and collaborationist Vichy government led by Marshal Philippe Pétain replaced the Third Republic following this agreement.
The 7th Panzer Division raced ahead refusing to allow rest while advancing day and night. This division advanced 40 kilometers in a single day under the command of Erwin Rommel who broke through French lines within 24 hours.
Italy entered the war on the 10th of June 1940 beginning the Italian invasion of France. After the armistice Italy occupied a small area south-east of France while the Vichy regime retained the free zone in the south.