Short answers, pulled from the story.
The Polish-Russian War was a conflict fought between the Polish-Lithuanian Commonwealth and the Tsardom of Russia from 1609 to 1618. King Sigismund III Vasa declared war hoping to gain territory, weaken Sweden's ally, and bring Orthodox Russia under Catholicism.
Poland exploited Russia's civil wars after the Rurik dynasty died out with Tsar Feodor I in 1598. A succession crisis and a famine from 1601 to 1603 weakened Russia, and Polish magnates backed pretenders to the throne known as the False Dmitrys before Sigismund III declared open war in 1609.
Yes, Polish troops entered Moscow in 1610 and garrisoned the Kremlin under commander Aleksander Gosiewski. They held it until a Russian siege forced their surrender, and the Kremlin fell on the 6th of November 1612.
Kuzma Minin was a respected town butcher of Nizhny Novgorod who oversaw the merchant funds, and Prince Dmitry Pozharsky led the military opposition. In 1611 they formed the Second Volunteer Army that drove the Polish occupation out of Moscow.
After the Poles were ousted from Moscow, the Russian Zemsky Sobor elected Michael Romanov, the 16-year-old son of Patriarch Filaret of Moscow, as tsar in March 1613. This inaugurated the Romanov dynasty, which ruled Russia for three centuries until the February Revolution in 1917.
The war ended in 1618 with the Truce of Deulino, signed on the 11th of December 1618. It granted the Polish-Lithuanian Commonwealth territorial concessions including Smolensk, Chernigov, and Severia, proclaimed a 15-year truce, and preserved Russia's independence.