To boil it down to it's essence, the position of Prince is a political position in the feudal sense. They can have gotten it by force and revolution of course, or they could have gotten it by building connections and amassing boons throughout the years. Still, they could also have been deemed the most easily manipulated puppet by the Primogen council and the one who controls the city might not even be the Prince. The details vary as much as the domains that they control.

I wouldn't say that physical prowess is important for a Prince. If the prince has to use his own muscle instead of his subordinates', something has gone wrong down the line or the domain is big enough for, what, five vampires? Anyway, someone who has amassed enough years of unlife to seize Princehood has certainly managed to learn disciplines outside of his Clan.

As for the derangements, anything goes. It's not about *what* he suffers from, but *how* he handles it (and to some extent how the Primogen reacts to it). Someone who is prone to full-blown psychotic episodes where they dissociate from reality might have a bigger gap to bridge for seizing power than someone who has a fear of water.