Short answers, pulled from the story.
Adoption in ancient Rome was primarily a legal procedure for transferring paternal power from one head of household to another. This process served financial, social, and political purposes among property-owning classes to perpetuate family estates and ancestral rites.
Adoptio involved the transferal of potestas over a free person from one head of household to another while adrogatio occurred when an adult acting sui iuris assumed the status of an unemancipated son for inheritance purposes. Adrogation of female adoptees became possible through imperial rescript in the Antonine era between AD 138 and 192.
Publius Clodius Pulcher surrendered his patrician status and became a nominal plebeian in order to qualify for the office of tribune during the late Republican era. Cicero criticized this adrogatio as solely politically motivated since Clodius was emancipated immediately after he had achieved his aim.
Augustus was adopted into the gens Julia in the will of his great uncle Julius Caesar on the 2nd of May 44 BC. Tiberius succeeded Augustus, Caligula followed Tiberius, and Nero Claudius Caesar succeeded Claudius as emperor Nero within the Nerva-Antonine dynasty united by a series of adoptions.
Roman women could own inherit and control property as citizens while playing an increasingly significant role in succession and inheritance from the 2nd century BC through the 2nd century AD. Testamentary adoption allowed women like Livia to become part of the Julian family upon her husband's death when she outlived him.