Leaking oil put of tail pipe, is the cooling sensor important, head gasket blown how do you know

Never seen oil come from tail pipe (unless a piston has a broken ring or a hole burned in it.) Cooling sensor tells the computer the engine temp and the computer turns the fan on through the fan relay. Very important for it to work. A blown head gasket will make the engine overheat, can cause white smoke from tailpipe, coolant loss and extremely rough running. Also can cause coolant in engine oil or oil in radiator. What you may be seeing may be water mixed with carbon from the inside of the pipe. Excessive water coming form pipe is also a head gasket symptom. A compression test or cylinder leakdown test or a radiator hydrocarbon test will tell you if gasket has failed.