Once you passed the population requirement and the first election took place, you'll always have a mayor. You can replace them, but that just means you get a _different_ mayor, who'll have the same requirements.
The room requirements of a mayor are significant but actually quite low compared with actual nobility. If you don't want to put in much effort, just make some fairly large rooms in stone, smooth the floor, install furniture of high quality and/or made from valuable materials. No need to break out the masterwork platinum furniture for your mayor, though, well-crafted brass is more than enough. If the room doesn't meet expectations, expand, engrave or throw in more/fancier furniture. Once you accept that 'decent' quarters take a _bit_ more effort than the 'meagre' 1x2 pigeonholes you cram your workers in, meeting the requirements is easy.
If the mayor already had personal rooms assigned before they got their new rank, take away those rooms (either 'f'ree the bed/chair/table or 'a'ssign to 'nobody'). Dwarfs with multiple rooms will use whichever room is nearest to them, and you don't want a mayor getting unhappy about 'very poor' lodgings, because they decided to sleep in their old meagre quarters instead of their mayoral great bedroom.
Edit: What mayors do - they're the chief administrator; as long as you have no nobility in your fort, the mayor will negotiate trade agreements with the dwarven liaison. Unhappy dwarfs will air their grievances with the mayor. And mayors can mandate the production of specific goods, and forbid their export. Much !FUN! can be had when a mayor orders the production of large gems or decides that the export of socks is an unacceptable practice.