There actually shouldn't be flow in a coldengine but if it's backing up, obviously, there is. The thermostat could be stuck open or it may be combustion pressure.
No, you don't need to see oil in the radiator or anywhere else to have a blown head gasket. It can be blown at many different points, the most common is between the cylinder and the cooling jacket. That would put compression into the coolings system. A pressure test on the cooling system with the spark plugs removed will expose that.