Short answers, pulled from the story.
Kublai Khan issued an edict titled Proclamation of the Dynastic Name that officially established the Great Yuan dynasty in 1271. The conquest was not complete until 1279 when Southern Song forces were defeated at the Battle of Yamen.
Zhenjin died before Kublai in 1285 and Zhenjin's third son Temür Khan then succeeded the throne ruling from 1294 to 1307 with support from his mother Kökejin and minister Bayan. Tugh Temür won civil war against Ragibagh in 1328 but Kusala died four days after banquet supposedly poisoned by El Temür who then remounted the throne as puppet ruler.
The population was divided into four classes based on loyalty and surrender date rather than pure ethnicity. Mongols held top position called Gao-chen meaning citizens of ruling empire while Nan or Southerners comprised all subjects of former Southern Song dynasty including ethnic Han minority groups in southern China and people of Dali Kingdom sat at bottom of privilege ladder sometimes called Manzi during Yuan dynasty.
Mathematician Zhu Shijie solved simultaneous equations with up to four unknowns using rectangular array coefficients equivalent to modern matrices. His method appeared in Jade Mirror of Four Unknowns written in 1303 containing diagram of Pascal triangle.
Red Turban Rebellion led Song loyalists started 1351 grew nationwide uprising establishing renewed Song dynasty Kaifeng capital 1351. From late 1340s countryside people suffered frequent natural disasters droughts floods resulting famines government lack effective policy led loss popular support.