The U.S. Constitution requires that state governments have a republican form of government. This requirement is found 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 you listed are not specifically mandated by the Constitution. States have the discretion to conduct elections as they see fit, including the structure of those elections. The Constitution does not require states to provide for the common defense (that is generally a federal responsibility), and there is no constitutional requirement for states to impose income taxes.