To answer your second and third questions - the dynastic names taken by imperial/royal Chinese states were usually selected because of a geographic connection to the ruling family or because of their auspicious connotations, the latter predominating from the Yuan on. This means that repeated dynastic names from early Chinese history often refer to unrelated regimes originating in similar places. Although complete fracturing did not happen as often after the Yuan, later examples from the sinosphere used regime titles more tactically and symbolically: the Manchu founder Nurhaci named his Manchurian regime the Jin dynasty (referred to historiographically as the Latter Jin) in a deliberate callback to the 12th century Jurchen Jin state, which had gone on to rule the North China Plain from a base in Manchuria.

Evelyn Rawski's book "Early Modern China and Northeast Asia: Cross-Border Perspectives" discusses this last example in more detail.