My race Z has an oil cooler. The oil is passed to it from an adapter on the oil filter boss on the engine, through an Oberg oil screen, then to the cooler and back to the adapter and the engine oil passages.
There is an oil temp sensor in the oil pan, and during a race the oil runs about 200 to 210 degrees f. The gauge reads from 140 to around 300. I hung some plates on it and took it for a drive to town once, and the oil temp barely made it to 150. When not running at full song, for sustained periods, the engine doesn't make enough waste heat to get the oil hot, so a thermostatic diverter valve would be needed if the car was driven casually more than in racing conditions.
Sure, the oil gets hot enough to be felt on the oil filter, on a street driven Z. In addition to lubricating the engine, it also absorbs some of the waste heat the coolant doesn't get, and it is dissipated somewhat through the oil pan. I bet if you took a non contact pyrometer and meqsured, the oil pan and filter housing might read around 160 to 170. Certainly not too hot considering the oil is changed frequently, i.e. every 5,000 miles or once yearly on a car driven infrequently.
As for the oil, I run 20-50 in the race car, and change it after every race meet. In my street Z 10-30 is fine, I'm only driving around town, not trying to be the first guy to the next stoplight. My 48 Ford 8N tractor gets the same 20-50 the race car gets, at 72 years old the flathead four engine still works hard. The Cummins diesel in my pickup, the diesel in my Kubota tractor and the Kubota mower all get Shell Rotella or Chevron Delo 15-40, oil formulated for diesel engines. My 2005 Subaru Legacy wagon calls for 5-30. Mrs. Racer has a 2012 Toyota Corolla, and it gets 0-10 oil, stuff is like water.
I don't get too fussy about brands, although Kendall GT1 is what I use in the race car.
And I never use additives. The oil manufacturers spend a lot of money and time figuring out the perfect addative package for each oil specification, and after it is consumed or otherwise altered from use, it is time to replace it.
Oil is cheap to insurance for expensive engines.