You likely gave this county to someone who was already a mayor. I remember that previously one of the counties in Ireland had a city as it's county capital; perhaps you granted the county to this person?
You can force the Lord Mayor to become a Count by finding his city holding and revoking it, so that he only has castle holdings. This will obviously piss him off, along with your other vassals.
Doges actually pay more taxes than feudal vassals do by default (as in they actually pay you taxes instead of nothing), so using them is a strategic choice. Similar with Prince-Bishops (although their use is less about money and more about stability). If you don't want either of those types of vassals, you will have to be sure not to grant feudal holdings to mayors or bishops (although bishops don't get the Wrong Government penalty). Also, if you manage to take control of a county that has either a city or bishopric as it's county seat, you'll either need to hand the county over to the local feudal baron, or revoke said baron to change the county seat to the castle.
