The U.S. Constitution requires that states must have a republican form of government. This is specified in Article IV, Section 4, which states: "The United States shall guarantee to every State in this Union a Republican Form of Government."
The other options listed are not mandated by the Constitution. While states conduct elections, there is no specific constitutional requirement regarding the types of elections (primary and general) that must be held. The Constitution also does not require states to provide for the common defense or to impose income taxes; those responsibilities are more broadly defined at the federal level, though states do have the authority to manage their own tax systems.