Short answers, pulled from the story.
Béla IV of Hungary fled his kingdom on the 11th of April 1241 after the Mongol horde under Batu Khan shattered the Hungarian forces at the Battle of Mohi. He escaped the carnage and was chased from town to town until seeking refuge in the coastal fortress of Trogir on the Adriatic Sea.
The Mongols withdrew from Hungary in March 1242 not because of Hungarian resistance but because the Great Khan Ögödei had died. This event forced Batu to return to the steppes for the succession, leaving Béla IV to face a country that had lost fifteen percent of its population to death and famine.
Béla IV of Hungary rebuilt his kingdom by allowing barons and prelates to erect nearly one hundred new stone fortresses across the kingdom. He also granted special liberties to thousands of colonists from the Holy Roman Empire and Poland to replace the lost population and persuaded the Cumans to return and resettle along the Tisza River.
The Cumans who fled to Hungary in 1239 committed robberies and rapes which created deep enmity between the king and his Hungarian subjects. The mob in Pest slaughtered Köten's retinue after the death of their leader, causing the Cumans to flee and destroy villages on their way to the Balkans.
The civil war between Béla IV of Hungary and his son Stephen ended with the Peace of Pressburg signed on the 23rd of March 1266. This agreement established a dual monarchy where the lands to the west remained under Béla's direct rule while Stephen governed the eastern territories.