Short answers, pulled from the story.
The Russian Provisional Government was led first by Prince Georgy Lvov and then by Alexander Kerensky, who became Prime Minister on the 21st of July 1917. Lvov resigned following the Kadet walkout and the July crisis.
The Russian Provisional Government lasted approximately eight months, from its formation on the 15th of March 1917 to the Bolshevik seizure of the Winter Palace on the 26th of October 1917.
The Russian Provisional Government failed because of political factionalism, an inability to enforce its own legislation, the disastrous June Offensive, and the growing power of the Petrograd Soviet, which controlled the army and railways. Scholar Pavel Osinsky described the Bolshevik takeover as 'socialism by default' rather than 'socialism by design,' attributing it as much to the government's failures as to Bolshevik strength.
The Petrograd Soviet operated as a rival power center that controlled the army and the railways, giving it the practical means to enforce policy that the Provisional Government lacked. Minister of War Alexander Guchkov acknowledged that the government had only 'the appearance of authority,' while real power rested with the Soviet.
In September 1917, General Lavr Kornilov, commander-in-chief of the Russian army, advanced troops toward Petrograd in an attempted coup. Kerensky branded Kornilov's forces counter-revolutionaries and ordered their arrest, which the army viewed as a betrayal of Kornilov, making it finally disloyal to the Provisional Government. Kerensky was forced to arm the Bolshevik Red Guard to stop Kornilov, which strengthened the Bolsheviks at the government's expense.
The Russian Republic was proclaimed on the 1st of September 1917 (the 14th of September, N.S.) in a decree signed by Alexander Kerensky as Minister-President and Alexander Zarudny as Minister of Justice. The proclamation came immediately after the suppression of the Kornilov coup attempt.