Both head gasket and trans cooler could be the problem. I think there is also an oil cooler tank built into the radiator which could explain oil in the coolant. Try doing cooling system pressure test, if you don't have the gear just visit your friendly radiator repair service - they will surely have the tool. Or you might want to invest about $60-70 and get your own from for example Dxxk Smith - 15Pcs Radiator Pressure Test Set Cooling System Leaking Check Refill Vacuum Tool.
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