Short answers, pulled from the story.
Adoption in ancient Rome was primarily a legal mechanism for transferring paternal power, called potestas, to ensure succession of a family's name, wealth, and ancestral religious rites. Cicero identified these three continuities, the nomen, pecunia, and sacra, as the core concerns adoptio was designed to protect. It was practiced mainly among the property-owning classes for financial, social, and political purposes.
Roman adoptio was not designed to build emotionally satisfying families or support childrearing. It was a legal transfer of paternal potestas between male heads of household, primarily used to secure a male heir and preserve a family's name, estate, and religious rites. Childless couples who simply wanted to raise children fostered instead of adopting.
Adrogatio was a form of Roman adoption in which the adoptee was already a legal adult acting independently, with no living father. Rather than one paterfamilias surrendering potestas to another, adrogatio merged two family lines. It required approval by the comitia calata and thirty magisterial lictors summoned by the Pontifex Maximus, and could only occur in the city of Rome until the reign of Diocletian.
Augustus was adopted posthumously by Julius Caesar through his will. Tiberius was adopted by Augustus, and Germanicus was adopted by Tiberius as a condition of that arrangement. The Nerva-Antonine dynasty produced five successive adoptive emperors: Trajan, Hadrian, Antoninus Pius, Lucius Verus, and Marcus Aurelius. Machiavelli called these the Five Good Emperors and attributed their success to having been chosen for the role rather than inheriting it by birth.
Adrogation of female adoptees became possible through imperial rescript in the Antonine era, from AD 138 to 192. In one documented case from the 3rd century, a woman whose sons had died was permitted to adopt her stepson. Women could not transfer paternal potestas, however, so formal adoption accomplished little for them that could not be achieved through inheritance law.
Publius Clodius Pulcher used adrogatio to surrender his patrician status and become a plebeian, qualifying him for the office of tribune, which was closed to patricians. Cicero criticized the act as purely political, and Clodius was emancipated immediately after achieving his aim. Publius Cornelius Dolabella followed the same path, and Augustan legislation later created additional incentives for sham adoptions by granting privileges to fathers of multiple children.