Kusha and Lava were twin sons of Lord Ram and Mata Sita. When their mother was pregnant, she had to leave her seat as a queen and go, live in Great Sage Valmiki’s Ashram at banks of Tamasa river as people were not able to tolerate that their Queen was confined to pleasure garden of Ravana named Ashoka Vatika for one year.
She gave birth to Kusha and Lava in the Ashram and they were educated by Maharishi Valmiki. They became great warriors and singers at a very young age, proficient in all Vedas.
When Shri Ram was doing his Ashwamedha Yajna, Kusha and Lava were called as they were singing Ramayana on the streets. Shri Ram told Bharat that the story will not be a pleasant one to hear (due to his exile and Sita’s abduction).
While singing the epic, they revealed to the King that they were his sons. Shri Ram was overjoyed to see his sons and he called Maharishi and Mata Sita. Mata Sita preferred to go back to Mother Earth than returning as Queen. While others praised her, Lord Ram’s looked agitated and disturbed. His mind was never at peace. He kept ruling and always placed Sita’s golden idol everytime he did Yajnas. Surprisingly, Kusha and Lava’s reactions are not mentioned in the Valmiki Ramayana. Kusha was given Kushavati and Lava was given Sravasti to rule.
Source- Valmiki Ramayana
The story of Kusha and Lava fighting with their father and uncles is in Padma Purana and other versions of Ramayana. Original Valmiki Ramayana does not mention his sons fighting him and to be honest, Shri Ram of Valmiki Ramayana would never tolerate this rebellious nature. Shri Ram of Valmiki Ramayana is very different from Shri Ram of Ramcharitmans.
Kusha and Champika
While Shri Ram was ruling Ayodhya, he got invitation of Swayamvar of the daughters of Maharaja Bhurikirti (King of eastern lands) named Champika and Sumati. He took his sons along with his brothers and their sons to the Swayamvara.
Champika chose Kusha as her husband and both kings were overjoyed and the marriage happened with pomp and grandeur.
Kusha had daughters with Champika.
Source- Ananda Ramayana
Lava and Sumati
Then Sumati decided to marry Lava and she chose him as her husband and it gladdened their fathers.
Source- Ananda Ramayana
Kusha and Kumudvati
Long after Shri Ram had given up his body, one night goddess of Ayodhya visited Kusha and asked Kusha to rebuild Ayodhya to its former glory. Here is the only hint in any sources available of how Kusha felt on the death of her mother because he was reluctant to go there where his mother didn't get the life she deserved. But he eventually went there with his ministers and people and with absolute hard work , he rebuild the city.
One day, he went on a pleasure tour in Sarayu river with his wives and servants. They all looked happy playing in the water and he decided to join them. When he came out he saw one of his golden bracelets gifted to him by Shri Ram was missing. He got angry because it was memory of his father and it got lost. Suddenly Naga King Kumud along with his beautiful sister Kumudvati came out of water. She was timid, shy and held the bracelet in her hand.
Kumud said his sister got the bracelet and she was playing with it. He returns the bracelet and gives his sister’s hand in marriage to Kusha. Kusha gladly accepted her hand and they got married right there and devas blessed the union. Kumud got safety from Garuda, vahan of Vishnu due to this marriage alliance.
Kumudvati soon became his favourite queen and he raised her to the dignity of Queen Consort. They eventually got a son named Atithi who was jewel of both dynasties. Kusha made him Prince regent and arranged his marriage to princesses. When he left to fight a giant named Durjaya, he corwned him as Yuvraj officially. He killed the giant but got fatally injured. When he died, Kumudvati followed him in death. They reunited in heaven where he shared half of the throne with Indra and she with sachi.
Source- Raghuvansh by Kalidas
Kalidas always exaggerated his stories but he was remarkably faithful to the names of the people, so history wise, quite accurate even if he added mythical, supernatural and romantic elements.