I would like to understand the status of the Nasi in the times of the talmud. It seems that this was both an inherited position as well as something that could be lost/taken away and given to another person. Did the Nasi have the halachic status of a king? If not what were the differences?
From these lectures by R' Ari Lamm it seems that there are two distinct periods where the title Nasi was in effect but the role was different. Before the destruction of the temple the Nasi was in some way affiliated with, if not outright the head of, the Sanhedrin. After the destruction and the Sanhedrin dissolved the Nasi was given certain communal authorities including the ability to judge and perhaps administer a death sentence. In the second lecture he references a source that the role of Nasi can be traced back to being a descendant of Dovid HaMelech but calls that source into question.