Short answers, pulled from the story.
A gens was a family group in ancient Rome made up of individuals who shared the same nomen gentilicium and claimed descent from a common ancestor. A gens could range from a single family to hundreds of individuals, and membership shaped a Roman citizen's social standing, religious duties, and legal rights.
A gens was the larger family grouping identified by a shared nomen gentilicium, while a stirps was a distinct branch within a gens, often identified by its own cognomen. The Tullii Cicerones, for example, were a stirps within the Tullia gens.
A Roman citizen's nomen gentilicium announced membership in a gens directly. In the name Marcus Tullius Cicero, Tullius is the nomen identifying him as a member of the Tullia gens, while Cicero identifies his specific branch. Women inherited their father's nomen in its feminine form, as with Cicero's daughter Tullia.
The gentes maiores and gentes minores were two tiers within the patrician gentes. The maiores were likely the oldest patrician families, probably including the Aemilii, Claudii, Cornelii, Fabii, Manlii, and Valerii. No definitive list survives, and the practical significance of the distinction remains uncertain.
The Claudii were admitted to the patriciate in 504 BC, five years after the establishment of the Roman Republic. This is recorded as the last definite instance of a gens being admitted to the patriciate before the first century BC.
Freed slaves, called liberti, typically assumed the nomen of the person who manumitted them, and sometimes also that person's praenomen. They were not technically full members of the gens whose name they took, but within a few generations their descendants often became indistinguishable from those with ancient roots in that family.