I have a few questions about Carthage:
Who led the state, and how were they chosen?
Was there anything similar to the Roman senate?
Did Carthage directly administer areas in Spain, Sicily and the North African coast or were these areas more like client states?
Given that nobody else has answered yet, I'll have a shot.
First a governor (skn) responsible to the King of Tyre, later one or more kings (mlk), and finally one, and later two, 'judges' (suffetes/sptm) elected on the basis of their nobility
Yes, a senate of noblemen appointed for life.
At most they were colonies, otherwise client-states or less. Carthaginian influence was often a nebulous thing in these areas.