Q163. "In Article IV section 4 it states 'The United States shall guarantee to every State in this Union a Republican Form of Government,' however it never states what that means exactly. Could a state have a parliamentary form of government, with an executive who is one of the legislators? Is a constitution even required?"
A. You are correct that a lot of different forms of government could qualify under the moniker "republican." In the context of the Constitution, "republican" basically means any form where the people choose their leaders. There is no requirement for a constitution, for example. A parliament would be fine, as would an executive that came from or had some role in the legislature. That would be up to the people of the state.
Of course at this point, all state legislatures and executives mirror, roughly, that of the federal government, with a bicameral legislature (excepting Nebraska, which has a unicameral legislature), and a separate executive. All states also have a constitution. The question is not just academic, however, because though the status quo tends to prevail, there is room for change.