Short answers, pulled from the story.
The term Tathāgatagarbha splits into two roots where tathagata refers to the one thus gone and points directly to Buddhahood while garbha translates as root, embryo, or womb. This combination creates a concept where every sentient being contains the potential for enlightenment within their own existence.
Brian Edward Brown dates the composition of the Śrīmālādevī Siñhanāda Sūtra to the 3rd century CE under the Andhra Ikshvaku dynasty. This text emerged as a product of the Mahāsāñghikas of the Andhra region who were responsible for the inception of this doctrine.
Scholars associate some of the earliest Tathāgatagarbha sūtras with specific early Buddhist schools in ancient India such as the Mahāsāñghikas of the Andhra region. Paul Williams writes that this collection appears less prominent in India compared to other traditions but became increasingly popular in Central Asian Buddhism and East Asian Buddhism instead.
The Lañkāvatāra Sūtra from the 3rd century CE synthesizes tathāgatagarbha with the emptiness of prajñāpāramitā sutras. It identifies the luminous mind of the canon with the tathāgatagarbha while stating that it might be mistaken for a self which it is not.
The Ratnagotravibhāga stands as the only Indian attempt to create a coherent philosophical model based on ideas found in the Tathāgatagarbha Sutras. Despite East Asian Buddhism's propensity for these concepts, the treatise has played a relatively small role there due to the primacy of sutra study.