Short answers, pulled from the story.
Temujin became Genghis Khan in 1206, uniting warring Mongol tribes into a single force. This unification followed fifteen years of unusually wet and warm weather from 1211 to 1225 that allowed horses to breed in unprecedented numbers.
Nearly half a million deaths occurred within a single army of perhaps two tumens after the fall of Urgench. Each soldier had a quota to execute twenty-four people, resulting in total destruction for resisting cities in Central Asia.
The Muslim Bahri Mamluks halted the Mongol advance at the Battle of Ain Jalut in southeastern Galilee on the 4th of April 1260. Most of the Mamluk military was composed of Kipchaks who had fled the initial invasions and were replenished by the Golden Horde.
Kublai Khan established the Yuan dynasty in China in 1271 as his greatest triumph. The Song dynasty fell finally in 1279 after decades of fighting following the subjugation of the Western Xia in 1209.
The Mongols withdrew from Eastern Europe after the death of Ögedei Khan in 1241. Their retreat allowed surviving states like Poland and Hungary to rebuild their defenses against future threats.