Did Lord Krishna have children? If yes, what were their names? What did happen to them after Krishna's death?
Yes, Lord Krishna had many children. As per Shrimad Bhagavatam the number was at least 161,080. Regarding His sons and grandsons, the scripture says their numbers were in millions:
My dear King, the sons and grandsons of Lord Kṛṣṇa’s children numbered in the tens of millions. Sixteen thousand mothers gave rise to this dynasty. [SB - 10.61.19]
Shri Krishna had 16,108 wives and He had 10 children from each. Of His wives, 8 were principal and all their children's names are mentioned in the Bhagavatam. They are as below:
The first son of Queen Rukmiṇī was Pradyumna, and also born of her were Cārudeṣṇa, Sudeṣṇa and the powerful Cārudeha, along with Sucāru, Cārugupta, Bhadracāru, Cārucandra, Vicāru and Cāru, the tenth. None of these sons of Lord Hari was less than his father. [SB - 10.61.11]
The ten sons of Satyabhāmā were Bhānu, Subhānu, Svarbhānu, Prabhānu, Bhānumān, Candrabhānu, Bṛhadbhānu, Atibhānu (the eighth), Śrībhānu and Pratibhānu.[SB - 10.61.12]
Sāmba, Sumitra, Purujit, Śatajit, Sahasrajit, Vijaya, Citraketu, Vasumān, Draviḍa and Kratu were the sons of Jāmbavatī. These ten, headed by Sāmba, were their father’s favorites. [SB - 10.61.12]
The sons of Nāgnajitī were Vīra, Candra, Aśvasena, Citragu, Vegavān, Vṛṣa, Āma, Śaṅku, Vasu and the opulent Kunti. [SB - 10.61.13]
Śruta, Kavi, Vṛṣa, Vīra, Subāhu, Bhadra, Śānti, Darśa and Pūrṇamāsa were sons of Kālindī. Her youngest son was Somaka. [SB - 10.61.14]
Mādrā’s sons were Praghoṣa, Gātravān, Siṁha, Bala, Prabala, Ūrdhaga, Mahāśakti, Saha, Oja and Aparājita. [SB - 10.61.15]
Mitravindā’s sons were Vṛka, Harṣa, Anila, Gṛdhra, Vardhana, Unnāda, Mahāṁsa, Pāvana, Vahni and Kṣudhi. [SB - 10.61.16]
Saṅgrāmajit, Bṛhatsena, Śūra, Praharaṇa, Arijit, Jaya and Subhadra were the sons of Bhadrā, together with Vāma, Āyur and Satyaka. [SB - 10.61.17]
Apart from these eight queens, Shri Krishna had 16,100 other queens and each had 10 sons each:
tāsāṁ yā daśa-putrāṇāṁ kṛṣṇa-strīṇāṁ puroditāḥ [SB - 10.61.7]
-Among Lord Kṛṣṇa’s wives, each of whom had ten sons
So if we calculate thus, then the number would be around 161,080.
Almost all of these children and their grandchildren died due to the quarrel and fighting that ensued among them in the Prabhasa region and due to the submerging of Dwaraka under the sea. But the great-grandson of Shri Krishna, Vajra, the son of Anirudha (Anirudha was son of Pradyumna ) survived and was appointed as King by Arjuna as described in Mahabharata.
The details of Lord Krishna's family and His offspring from ashtabharya's [eight principal queen/consorts]
List of offspring classified as patrani's or ashtabharya's of Lord Krishna
Pradyumna, Charudeshna,Sudeshna, Charudhena, Sucharu, Chharugupta, Bhadracaru, Charuchandra, Vicaru and Caru.
Bhanu, Subhanu, Svarbhanu, Prabhanu, Bhanuman, Chandrabhanu, Brihadbhanu, Atibhanu, Sribhanu and Pratibhanu.
Samba,Sumitra, Purujit, Satajit, Sahasrajit, Vijaya, Citraketu, Vasuman, Dravida and Kratu.
Vira, Candra, Asvasena, Citragu, Vegavan, Vrisha, Ama, Sanku, Vasu ,Kunti.
Sruta, Kavi, Vrisha, Vira, Subahu, Bhadra, Santi, Darsa ,Purnamasa.
Vrika, Harsha, Anila, Gridhra, Vardhana, Unnada, Mahamsa, Pavana, Vahni and Kshudhi.
Sangramajit, Brihatsena, Sura, Praharana, Arijith, Jaya and Subhadra.
Praghosha, Gatravan, Simha, Bala, Prabala, Urdhaga, Mahasakti, Saha, Oja and Aparajita.
Because of the unrighteous behaviour of Samba (the son of Jambavanti) all unrighteous Yadava's were wiped out from earth,Lord Krishna had warned them about this massive wipe out (Yadavas wiped out).
Lord Krishna said he preformed severe austerities to obtain Pradyumna as his son.
That son whom I obtained through ascetic penances and observances of austere brahmacarya for twelve years on the breast of Himavati whither I had gone for the purpose, that son of mine, Pradyumna, of great energy and a portion of Sanat-kumara himself, begotten by me upon my wife Rukmini who had practised vows as austere as mine, that hero even never solicited this best of objects, this unrivalled discus, which thou of little understanding had solicited!
He did penance to obtain Samba also.
"The blessed Krishna said, 'Bowing my head with great joy unto that mass of energy and effulgence, I said these words unto the great Deity, with a heart filled with gladness,--Firmness in virtue, the slaughter of foes in battle, the highest fame, the greatest might, devotion to Yoga, thy adjacence, and hundreds upon hundreds of children, these are the boons I solicit of thee,--So be it,--said Sankara repeating the words I had uttered. After this, the mother of the universe, the upholders of all things, who cleanses, all things, viz., the spouse of Sarva, that vast receptacle of penances said with a restrained soul these words unto me,--'The puissant Mahadeva has granted thee, O sinless one, a son who shall be named Samva.