Not sure I can add anything that might help with your tuning issues, but I'll try.
Apologies in advance if I am repeating other posts, man this is a long thread....
First, closed vs open loop .
Generally speaking, when the ECU is in closed loop it is using the O2 sensor to adjust the air/fuel ratio.
When in open loop, it still can see the O2 sensor, but it doesn't do anything with that info.
That's about the long and short of it.
Open loop is there to give the sensor time to heat up ( it needs to be around 750C to operate ) and also because you purposely want the engine rich on cold starts.
Now obviously, a self tuning system can only self tune when it is in closed loop, that's when it knows it's getting valid info from the O2. During open loop, it's still a trial and error thing where you have to tell it to go richer or leaner, think adjusting the choke ona carb.
If there were a "self tuning" carb that used a modern computer, the computer would dial in the jets for you, but you would still need to pop the air cleaner and twist the choke stove to get the cold starts just right,if that makes any sense.
So, the fact that your engine runs fine once it goes into closed loop tells me that your air fuel ratio is off. I'm not familiar with your system, so I don't know what parameters you have control over.
But I would look for a cold start, after start, cold compensation, warmup enrichment or choke type setting, and try leaning it out. If it gets worse, then fatten it up.
I can speak from a lot of experience here, cold stars are probably one of the most frustrating things to tune on a fuel injected motor, because you get one shot at it, then the engine is warm and doesn't act the same way on subsistent partially warm starts. You have to let it cool all the way down and try again.
As far as the O2 getting damaged by oil, generally is got to be a massive oil burner to hurt the O2, the element inside sensor is heated and glows red hot within about 30 seconds of start up, so it does a pretty good job of burning off what oil might get on it. Now coolant in the exhaust is another matter, that will kill an O2. So my advice is, if your O2 sensor readings look good, your sensor is probably just fine.
More advanced systems will let you add or subtract ignition timing based on engine temp, not sure yours lets you do this, but that generally only adds or subtracts a few degrees either way, more advance when it's cold, might retard it a few degrees if it gets really hot. From what you describe i doubt it has anything to do with temp related timing adjustments.
Not sure if this helps, or is even leading you in the right direction, but I hope it does.
Joe
