No, I don't believe your engine has an oil cooler.
This engine is notorious for oil leaks at the timing chain cover at the front of the head, and at the right rear corner (passenger side) of the head.
2001 E430, Bourdeaux Red, Oyster interior.
1973 280SE 4.5, 170,000 miles. 568 Signal Red, Black MB Tex. "The Red Baron".