The Mongol empire apparently lasted until 1368. What happened at this point and after this? Did amyone take any kind of revenge on them considering the magnitude of the suffering they inflicted on peoples across the world?
There's a couple of answers to this question so I'll leave the far east to someone else but it we talk about what happened in the near East - the Mongols never really left as, rather than shrinking, the empire disintegrated into a multitude of smaller states, partly due to the impact of the Black Death making travel across Asia exceptionally difficult cutting off communications and breaking ties. These smaller Khanates lived on for much longer than 1368 - some carrying on into even the 18th Century.
Tamerlane is the best example of this having descended from Mongols and come from a Mongol tribe but calling himself the Sword of Islam, he became the most powerful man in the Muslim world, killing 1/20th of the world's population as part of his campaigns. The 1368 date you refer to is simply when the Mongols were driven out of China - remnants of their empire persisted for far longer due in no small part to the exceptional ability of parts of Mongol society to adapt to local conditions as seen at it's finest in the Muslim world
On the contrary, being a descendant of Genghis Khan was seen as something that legitimised a Dynasty or a regime. Amir Temur, better known as Tamerlane kept a puppet Mongol Khan to legitimise his rule * . Out of the four great Muslim empires of early modernity two were able to trace their descent from Mongol royalty; namely the Mughals and Özbeks. In fact, the Ottoman Empire had some issues with its self image as it was the only one out of those four that couldn't trace their history back to such a legitimising factor**.
Sources: Finkel, Osman's Dream & Marozzi, Tamerlane.