Can you overcool the oil?
Are you guys running the remote filter set up or the adapter that allows you to keep the filter on the frame?
Yes you can. Don't confuse water temp with oil temp. They are not the same and only slightly related.
If you look at people who run oil coolers on street engines you will often see duct tape across the cooler when it is cold outside or just running around on the street. that is to keep the oil from over cooling.
For a street application there are only 2 ways I would ever consider running an oil cooler. First is an oil to water cooler. I really prefer these. It will help bring the oil up to temperture when it is cold out. Generally plumbed into the hot side of the raditaor tank if done in radiator. The stock truck oil coolers are plumbed entirely into the lower radiator hose, which probably works with the reduced flow of the plumbing but you loose the benifit of heating the oil with the water.
If all you plan to do is an oil to air cooler (which will take the stress off the water radiator system) then I would only consider it if running an oil temperture thermostat, This bypasses the cooler when cold to give the oil a chance to warm up.
The correct temperture isn't "as cold as possible", it is "normal operating temperture". Now the debate where normal operating temperture is open. I like high 100's, 180, 190. Some low 200s are good too, drives off the moisture. Mid 200s is getting warmer than I like. I have seen 285 once, stupid Chrysler put an oil temp gauge in a 300. If you hold 6000 RPM for 10 minutes or so the oil will get really hot;D. I also don't like really cold oil, if you can't keep it above 150 it is too cold. Cold oil has good oil pressure, but oil pressure is the resisitance to flow. I would rather warm the oil, loose a little pressure and know I have good flow. Cold oil can't drive out the moisture and will become contaminated in short order.