I know that there were mayors in Paris and Montpelier. I'm guessing the larger communities had mayors and smaller ones didn't. So probably Bloomington, Georgetown, and St. Charles also had mayors, but Bennington, Bern, Dingle, Fish Haven, Geneva, Liberty, Nounan, Ovid, Pegram, and Wardboro probably didn't. I'll see if I can't find confirmation in some local histories. That being said, during the early decades the mayors probably didn't have much power. Many government functions were performed by LDS Church officials during that time period.
The people did have voting rights, except during period of the "Test Oath" from 1884 to 1892. The oath was intended to prevent Mormons from voting. Basically if you were Mormon, you couldn't vote or hold public office.The test oath was actually became part of the Idaho State Constitution and although it hadn't been enforced for decades, it wasn't actually removed until 1982.
I asked my mother, Dora Price Larsen, if she knew of Mayors in Bloomington. She said she was not aware of anyone designated as Mayor but they had "City Dads". Her father, Joseph Stanners Price was a city dad who helped develop the water works for Bloomington.