Can a mayor, town official, or other official order the flag to be displayed at half-staff?

The answer is NO, not legitimately

The solemn practice of placing the American Flag at half-staff means that our Nation or the state is mourning the death of a highly regarded National or state figure. Therefore only the President of the United States or the Governor of the state may order the Flag to be half-staffed in accordance with Flag Code section 7(m).

Thus, when you encounter mayors or town officials who take the Flag Code into their own hands it must be recognized that these individuals are acting against the US Flag Code. These actions are also minimizing the acts taken by governors or the President.