I bought a home in the Hilliard school district. My address was Hilliard, OH 43026. My street sign was green. Three streets over, Hilliard-proper street signs are blue. My trash was picked up by the city of Columbus, Hilliard's by Rumpke. If I called 911 Columbus Fire and/or police responded. For Hilliard proper it was Norwich township Fire and Hilliard PD. My phone prefix was 527, a Hilliard number. My taxes were about 20% lower than folks on blue-signed streets, but so was my property valuation by the county. But my son went to Hilliard schools, and that's all that matters.
The suburbs all have elected mayors and city councils. But they are limited in big ways, as Columbus has its tentacles into so much infrastructure in the county. Water and sewer is a big case in point.
As for Franklin County, their biggest issues cover the courts, metro parks and some law enforcement (sheriff) as well as the oddball stuff like the Airport Authority which controls Port Columbus, Rickenbacker and Bolton.