Short answers, pulled from the story.
The Allied invasion of Italy began on the 3rd of September 1943, when British forces crossed from Sicily into Calabria in Operation Baytown. The main assault, Operation Avalanche, landed at Salerno on the 9th of September 1943, alongside a supporting landing at Taranto in Operation Slapstick.
General Sir Harold Alexander commanded the overall 15th Army Group during the Allied invasion of Italy. Beneath him, Lieutenant General Mark W. Clark led the U.S. Fifth Army at Salerno and General Bernard Montgomery led the British Eighth Army in Calabria.
The Allies invaded Italy to remove it from the Axis, weaken German influence in the Mediterranean, and tie down German forces that could otherwise fight elsewhere. Winston Churchill had called Italy the soft underbelly of the axis, though Mark Clark warned it would prove to be one tough gut.
At Salerno, the U.S. 36th Infantry Division and British 46th and 56th Divisions landed on the 9th of September 1943 and met strong German resistance. A German counterattack on the 13th of September nearly broke through the Allied line before paratroopers of the 82nd Airborne Division reinforced the beachhead and the line held.
The Salerno mutiny occurred on the 16th of September 1943, when about 500 men of the British X Corps refused orders assigning them to unfamiliar units as battle casualty replacements. Three NCOs identified as ringleaders were sentenced to death, though the sentence was never carried out.
Naples fell to the Allies on the 1st of October 1943, when "A" Squadron of the 1st King's Dragoon Guards became the first Allied unit to enter the city. The Fifth Army reached the Volturno River by the 6th of October 1943, securing the surrounding plain.