Short answers, pulled from the story.
A polity is a group of people with a collective identity, organized by political and institutionalized social relations, with a capacity to mobilize resources. It is the basic unit of a political community or body politic. Polities include states, nations, empires, provinces, and international organizations.
No. A polity like a state does not need to be a sovereign unit. Dependent territories, constituent countries within a larger sovereign state, and states with limited recognition all qualify as polities.
The preeminent polities today are Westphalian states and nation-states, commonly referred to as countries. The term country may refer to a sovereign state, a state with limited recognition, a constituent country of a sovereign state, or a dependent territory.
Thomas Hobbes was a highly significant figure in the conceptualisation of polities, in particular of states. He considered notions of the state and the body politic in Leviathan, his most notable work.
A polity may encapsulate a multitude of organizations. Many of these form the administrative apparatus of contemporary nation-states, including subordinate civil, regional, and local government authorities.
In geopolitics, a polity can manifest as a province, a nation, a state, an empire, an international organization, a political organization, or another identifiable, resource-manipulating organizational structure.