Yep
You have to define "trace" - glycol is usually expressed as either present, not, or "oh $hit you're screwed".
The latter doesn't require an oil analysis to see.
The oil cooler core CAN be the cause of this, but its not terribly likely if this is an isolated issue with nothing else going on.
I'd pull the oil cooler before I pulled the head, to be honest. Its easier and YOU can do it. However, make damn sure you can get all the gaskets first, because without them its not going back together. You will also need someone's help to test it - the proper procedure requires making up a blanking plate so you can put the oil side under shop air pressure (~50-60lbs or so) and then immersing it in water to see if you get bubbles. There is a risk of rupture when you do this, so care is required.
BUT - in nearly ALL cases if you have a bad oil cooler core you also have oil in the coolant. Open up the neck in your tank and look. Oil floats, so it'll be VERY visible. The reason is that oil pressure is vastly higher than coolant pressure - so the leak is in that direction. When you shut down coolant will get in the oil, but its basically impossible to get that without major cooling system contamination.
Hope you DO find that, by the way, although it too is a royal pain - you need to flush the cooling system and run a de-oiling compound in there if it has gotten oil-contaminated - this is very, very important because oil in the cooling system will impede heat transfer enough to cause major trouble in the form of hotspots in the engine.
Wet liner engines are subject to severe damage if overheated - even slightly. The seals are just O-rings and if you "cook" them, even a bit, they will leak. That's a very, very common problem - the second being the head (cracked) or head gasket. The root cause of BOTH maladies is usually an overheat. My general "rule of thumb" is that if a wet liner engine is overheated and the head comes off (e.g. cracked, blown head gasket, etc) I give serious consideration to a full major, because it sucks to have to tear into it twice and there is no way to know for certain that the liner seals are ok after such an event.
Retorquing the head will usually not fix this kind of thing. Among other things unless it was done wrong originally or a bolt has failed due to a manufacturing defect the odds of trouble here are slim and none. Once the gasket fails it has to come off and be replaced most of the time anyway. Internal coolant leaks are extremely damaging - into the oil is bad, and into a cylinder is frequently worse, since its entirely possible to hydrolock a hole and if you do then you will snap the conrod when you go to crank up. That sort of failure can hose the block and leave you with a non-rebuildable engine.
