how did pre-Meiji "clans" work in Japan, e.g. Shimazu clan / Satsuma?

Factual Questions
1

I notice how political entities like Satsuma or Choshu are described as being ruled not just by a hereditary ruling family but by an institution called a “clan”. E.g. the clan ruling Choshu is described at having annual meetings where the “elders” ritually proposed to the ruler that they should rebel against Tokugawa.

Anyway, so how did these clans work? How did people become members? Could they quit and go join another clan? What percentage of the samurai living in a particular province belong to the local ruling clan?

2

The clan is the rest of the family who isn’t “the leader”. I believe that kids could be adopted and hence become part of the family. I’m not sure if there’s any other sort of points of entry.

3

Clans could also have cadet branches, of course.