Is there a Hadith that mentions the Dajjal's parents, and that they are believers, and that he claims righteousness, then prophethood, then divinity, and that he throws his mother into the Fire because of her anger at his claim to divinity? And what is the extent of its authenticity?

The Dajjal has existed since ancient times, as evidenced by the hadith (definition) of Tamim Al-Dari, which narrates his encounter with the Dajjal, who was imprisoned on an island, and the Prophet, peace and blessings be upon him, affirmed this hadith. As for what is narrated about the Dajjal's beginning by claiming righteousness, then prophethood, then divinity, it is a hadith with a weak chain of transmission and is not authentic.