Short answers, pulled from the story.
The Donghu confederation occupied eastern Mongolia and Manchuria in 699 BCE. This period marks one of the earliest recorded homelands for proto-Mongolic peoples according to Chinese histories.
Genghis Khan united Mongolic-speaking tribes into an umbrella term called Mongol beginning in the thirteenth century. The empire expanded from the Adriatic Sea to Indonesian Java and from Japan to Palestine during his reign.
Approximately 200,000 Kalmyks began migrating from their Volga pastures to Dzungaria in January 1771 but only 66,073 reached Balkhash Lake after seven months of travel. Empress Catherine the Great ordered Russian armies to exterminate the migrants while Kyrgyzs attacked them near Balkhash Lake where about 100,000 to 150,000 could not cross because the river did not freeze in winter 1771.
At least 4.1 million ethnic Mongols reside in China's Inner Mongolia Autonomous Region according to 2010 estimates. Across whole of China language is spoken by roughly half of country's 5.8 million ethnic Mongols though exact number remains unknown due to no data on language proficiency.
Gelug school of Tibetan Buddhism became state religion of Mongolia in 1576. The Red Hat school coexisted with Gelug Yellow Hat which was founded by half-Mongol Je Tsongkhapa between 1357 and 1419.