Cuda, the tranny is not in three sections. The tranny is the middle part that your talking about. That is the actual gearbox. The front part is the bellhousing. On some transmissions it is integrated into the casing but for the most part is a seperate piece. It does house the clutch and basically acts as a coupler between the engine and transmission. The last section is the tailshaft. The output for the transmission is there and thats where the driveshaft connects. There are several seals in the whole assembly that may or may not be leaking.

Have you changed the shifter for an aftermarket piece or anything of that nature?

Also, turndowns are the way to go to exit under the car. That will alleviate the carbon buildup on the underside. However, its a V6, swap an 8 in it or put the exhaust back on.