Did any Kaurava survive the Mahabharata war? If so, what happened to them after the war?
Lord Krishna talks about surviving warrior of Mahabharat war in this chapter of Mahabharat:
The Mahabharata, Book 14: Aswamedha Parva: Anugita Parva: Section LX Then, in thegreat battle that ensued, the royal son of Dhritarashtra was slain by Bhimasena who put forth his great prowess, in the presence of many kings. After this the remnant of the Pandava army, as it slept in the camp, was slaughtered at night time by Drona's son who was unable to put up with the slaughter of his father (at the hands of Dhrishtadyumna). Their sons slain, their forces slain, only the five sons of Pandu are alive with myself and Yuyudhana. With Kripa and the Bhoja prince Kritavarman, the son of Drona represents the unslain remnant of the Kaurava army. Dhritarashtra's son Yuyutsu also escaped slaughter in consequence of his having adopted the side of the Pandavas. Upon the slaughter of the Kaurava king (Suyodhana) with all his followers and allies, Vidura and Sanjaya have come to the presence of king Yudhishthira the just. Even thus did that battle occur, O lord, for eight and ten days. Many kings of Earth, slain therein, have ascended to Heaven.'
Hence Yuyutsu was the only Kaurav who survived Mahabharat war as he fought in side of Pandavas. All other 99 Kauravs were killed by Bhima.
But Yuyutsu wasn't son of Gandhari. He was son of Dhritarastra from another Vaisya wife named Sughada:
In the Sambhava Parva of Mahabharat:
"Janamejaya said, 'O sinless one, thou hast narrated to me from the beginning all about the birth of Dhritarashtra's hundred sons owing to the boon granted by the Rishi. But thou hast not told me as yet any particulars about the birth of the daughter. Thou hast merely said that over and above the hundred sons, there was another son named Yuyutsu begotten upon a Vaisya woman, and a daughter. The great Rishi Vyasa of immeasurable energy said unto the daughter of the king of Gandhara that she would become the mother of a hundred sons.
He took the side of Pandava as:
The Mahabharata, Book 6: Bhishma Parva: Bhagavat-Gita Parva: Section XLIII
Casting his eyes then upon them, Yuyutsu said these words, with a cheerful heart, unto Kunti's son king Yudhishthira the Just,--I will fight under thee in battle, for the sake of you all, with the sons of Dhritarashtra, if, O king, thou wilt accept me, sinless one.' "Yudhishthira said, 'Come, come, all of us will fight with thy foolish brothers. O Yuyutsu, both Vasudeva and we all say to thee--I accept thee, O thou of mighty arms, fight for my cause. On thee rests, it seems, the thread of Dhritarashtra's line as also his funeral cake. O prince, O thou of great splendour, accept us that accept thee. The wrathful Duryodhana of wicked understanding will cease to live.'"
Latter Yuyutsu served as Parikshit regent as described here:
The Mahabharata, Book 17: Mahaprasthanika Parva: Section 1 Bhimasena and the twins fully endorsed the words that Arjuna had said. Resolved to retire from the world for earning merit, they brought Yuyutsu before them. Yudhishthira made over the kingdom to the son of his uncle by his Vaisya wife. Installing Parikshit also on their throne, as king, the eldest brother of the Pandavas, filled with sorrow, addressed Subhadra, saying, ‘This son of thy son will be the king of the Kurus. The survivor of the Yadus, Vajra, has been made a king. Parikshit will rule in Hastinapura, while the Yadava prince, Vajra, will rule in Shakraprastha. He should be protected by thee. Never set thy heart on unrighteousness.’