Short answers, pulled from the story.
The Republic of China was proclaimed on the 1st of January 1912, following the Xinhai Revolution, which began with the Wuchang Uprising on the 10th of October 1911. Sun Yat-sen served as the provisional president, having been elected by representatives from seventeen provinces on the 29th of December 1911. He handed the presidency to Yuan Shikai shortly afterward.
The Kuomintang-led Nationalist government lost the Chinese Civil War to the Chinese Communist Party and its People's Liberation Army. Despite advantages in troops, weapons, and territory, the Nationalists were exhausted from the war with Japan, undermined by corruption and hyperinflation, and losing the battle for popular support, especially in the countryside. A few hundred thousand Nationalist troops and approximately two million refugees fled to Taiwan, and Chiang Kai-shek proclaimed Taipei the temporary capital on the 7th of December 1949.
The Nanjing decade refers to the period from 1927 to 1937, when Chiang Kai-shek's Nationalist government was based in Nanjing and China experienced relative stability. During this period the Academia Sinica and the Central Bank of China were founded, Chinese industries grew considerably, the highway network expanded from 1,000 km to 109,000 km, and the 1931 Civil Code granted women equal inheritance rights and the right to initiate divorce.
The war, which began in July 1937, was catastrophic. Half of the officers who graduated from the Central Military Academy were killed in the first few months of fighting. Over 100 of the navy's 120 ships were sunk or destroyed within the first two years. The ROC Air Force, which had 645 aircraft in 1936, was largely destroyed by December 1937. The war was estimated to have killed between 20 and 25 million Chinese in total, and by 1946 Chinese industries operated at only 20% of pre-war capacity.
The Republic of China was a founding member of the United Nations and held a permanent seat on the UN Security Council, representing China as one of the Big Four Allies from World War II. The ROC retained that permanent seat until 1971, when it was supplanted by the People's Republic of China.
The Warlord Era was triggered by the death of Yuan Shikai in 1916. Yuan had maintained control of the Beiyang Army by building personal loyalty between division commanders and their troops, which meant that after his death the army fragmented among rival generals rather than remaining under central authority. From 1916 onward, regional military commanders known as warlords controlled large portions of the country until the Kuomintang's Northern Expedition reunified China nominally in 1928.