Short answers, pulled from the story.
The Battle of France began on the 10th of May 1940 and ended with the signing of the Second Armistice at Compiegne on the 22nd of June 1940, a span of about six weeks. Germany had invaded Belgium, Luxembourg, the Netherlands, and parts of France, occupying Paris on the 14th of June.
The Manstein plan, formally known as Fall Gelb, proposed concentrating German armoured forces at Sedan in the Ardennes, breaking through the Allied line, and racing to the English Channel to encircle the Allied armies in Belgium. Generalleutnant Erich von Manstein wrote the first memorandum outlining this approach on the 31st of October 1939; Hitler ordered it adopted on the 18th of February 1940 after meeting Manstein directly.
Yes. France deployed 3,254 tanks on the north-eastern front on the 10th of May 1940, against 2,439 German tanks. French tanks generally had superior armour and armament, but most lacked radios, were slower, and were dispersed as infantry support rather than concentrated in offensive formations.
On the 13th of May 1940, Panzergruppe Kleist forced three crossings of the Meuse near Sedan using the 1st, 2nd, and 10th Panzer divisions. The Luftwaffe flew 3,940 sorties in the heaviest air bombardment the world had yet witnessed to suppress French defences. A panic known as the "Panic of Bulson" caused the 55th Infantry Division to abandon its position twelve hours before German tanks actually arrived, opening a gap that led to the collapse of the French line.
The Maginot Line was not breached directly; it was outflanked. The line ran from the Swiss border to Longwy and was intended to channel a German attack into Belgium. German forces instead drove through the Ardennes to the north of the line, crossed the Meuse at Sedan, and raced to the Channel, cutting off Allied armies before the fortifications could play any defensive role.
Churchill flew to Paris on the 16th of May 1940 after French Prime Minister Paul Reynaud telephoned him to say France was defeated. At a meeting with French commanders, Churchill asked General Gamelin where the strategic reserve was; Gamelin's reply that there was none was later described by Churchill as the most shocking moment of his life.