When Ward asks Eddie Haskell how long he's been driving Eddie replies, "In three weeks it will be two months". However, in the previous season Eddie drops Wally off in what June calls "Eddies car". Chronologically, this would mean Eddie's been driving for at least five months.
In all states, the agency responsible for issuing drivers' licenses is always run by the state, not individual cities. But, in Mayfield's case, its name is Mayfield Motor Vehicles Department. (This was done to avoid revealing which state the show is set in.)