To answer your first question, Yashoda's daughter was an incarnation of Yogamaya, the illusory power of Vishnu who is often associated with Durga; here is what Vishnu tells Yogamaya in this chapter of the Srimad Bhagavatam:
O all-auspicious Yoga-māyā, I shall then appear with My full six opulences as the son of Devakī, and you will appear as the daughter of mother Yaśodā, the queen of Mahārāja Nanda. By sacrifices of animals, ordinary human beings will worship you gorgeously, with various paraphernalia, because you are supreme in fulfilling the material desires of everyone. Lord Kṛṣṇa blessed Māyādevī by saying: In different places on the surface of the earth, people will give you different names, such as Durgā, Bhadrakālī, Vijayā, Vaiṣṇavī, Kumudā, Caṇḍikā, Kṛṣṇā, Mādhavī, Kanyakā, Māyā, Nārāyaṇī, Īśānī, Śāradā and Ambikā.
As far as what happened to her after Kamsa tries to kill her, she rose up into the heavens and then started residing in the Vindhya mountains, which is why she is known as Vindhyavasini; in this chapter of the Harivamsa, Vishnu tells Yogamaya what she'll do:
Catching your legs, he will try to smash you against a stone. You will slip out (of his hands), arise in the sky, and will attain an eternal place.
You will have the same complexion, dark, as mine. Your face will be similar to saMkarshaNa. In the sky you will appear with vastly spread arms like mine.
In your four arms, you will carry a trident, a sword with golden handle, a pot filled with honey (madhu) and a pure lotus.
You will wear blue silk and a yellow silk shawl. Your bosom will be adorned by a white necklace shining brightly like moon rays.
Your ears will be decorated with divine earrings. You will shine with your face, beautiful like moon.
You will shine with an exquisite crown and a beautiful hair style. With your vast arms beautiful like the hood of a serpent you will enchant the ten directions.
You will shine with a flag adorned with peacock feathers. You will dazzle with armlets decorated with peacock feathers.
Always surrounded by terrifying bhUtagaNa-s, you will follow my orders.
Taking a vow of celebacy (kaumAraM vrataM) you will go to heaven.
There the thousand eyed indra, along with deva-s will perform your abhiSheka, according to my order in the prescribed manner.
Then indra will formally accept you as his sister. As you are connected with the lineage of kushika, you will become known as kaushikI.
indra will allot you a permanent place in the best among the mountains, vindhya. Afterwards, you will glorify the earth with thousand installations.p
You will travel in all three worlds and any prayer to you will be fruitful. You will assume any form at your will and grant boons.
There, fixing me in your mind, you will destroy two mountain-living dAnava-s shumbha and nishumbha, along with their followers.
For those who don't know, Shumbha and Nishambha were two Asuras who took over the three worlds and were defeated by Ambika, a form of Durga, as I discuss in this answer. Apparently Vindhyavasini fought a rebirth of these demons, as described in this chapter of the Devi Mahatmya of the markandeya Purana:
40-41. The Devi said : 'When the twenty-eighth age has arrived during the period of Vaivasvata Manu, two other great asuras, Shumbha and Nishumbha will be born. Then born from the womb of Yasoda, in the home of cowherd Nanda, and dwelling on the Vindhya mountains, I will destroy them both.'
But I haven't found the details of this story, which is why I asked this question.