During takeoff, the pilot was distracted by inbound traffic and delayed retracting his gear. At the same time, the engines were not synchronized and he was having difficulty getting them to sound good. He then noticed that a fire had started in the vicinity of the left engine. The left engine was shut down and feathered, and the fire subsided. Since the hydraulic pump was on the left engine, the hand pump had to be used to retract or extend the landing gear. The pilot delayed pumping the gear up, then realized he would be unable to return to the airport. Subsequently, he landed in a grassy area with the gear retracted. An investigation revealed fire damage in the vicinity of the fuel injector and alternator. A fuel leak was found at the mixture control valve, PN 2522591, where the O-rings around the valve stem were worn. Also, a teardown of the alternator revealed that the brushes and armature were worn to the extent that sparking could occur. The density altitude was about 8500 ft.