Short answers, pulled from the story.
The Petrograd Soviet was established on the 27th of February 1917, the opening day of the February Revolution, as a representative body for Petrograd's workers and soldiers. Its founding followed the release of the Central Workers' Group leadership from the Peter and Paul Fortress, where they had been held since the 27th of January on orders from Interior Minister Alexander Protopopov.
Dvoevlastie, or dual power, was the condition in which the Petrograd Soviet and the Provisional Government simultaneously claimed authority over Russia in 1917. The Soviet created this condition by issuing its own orders, controlling infrastructure including telegraphs, railroads, and the food supply, while formally remaining outside the government and only conditionally supporting it.
Nikolay Chkheidze chaired the Soviet from the 12th of March 1917 to the 19th of September 1917. Leon Trotsky succeeded him, elected on the 25th of September 1917 after the Bolsheviks secured a majority in the Workers' Section. Grigory Zinoviev then served as chairman from the 13th of December 1917 through the 26th of March 1926.
The Bolsheviks gained a third of the vote in the August 20th municipal elections, a fifty percent increase over three months. They secured a majority in the Workers' Section on the 25th of September 1917, when Leon Trotsky was elected chairman and directed the Soviet's transformation into a Bolshevik revolutionary organ.
The Military Revolutionary Committee, or Milrevcom, was created by the Petrograd Soviet on the 16th of October 1917, officially to defend the city. Chaired by Pavel Lazimir with Nikolai Podvoisky as deputy, it served as a front for the Bolshevik Military Organization. On the 24th of October it directed the seizure of telegraph offices and bridges, and on the 25th issued Lenin's announcement declaring the end of the Provisional Government.
The Second Congress of Soviets dismissed the old Ispolkom on the 26th of October 1917 and replaced it with a new 101-member body, 62 of them Bolsheviks, under Lev Borisovich Kamenev. The Congress also approved the Decree on Peace, the Decree on Land, and created the Council of People's Commissars, or Sovnarkom, which in practice eclipsed the Central Executive Committee in power.