Give the coolant system a very good flush out, a lot here use wyns coolant flush.
dont be afraid to do it a few times. then add the 50/50 antifreeze mix keep a check on your filler cap they tend to play up. Use your own judgement you will know if the oil cooler gives up. Check further weekly for water or oil loss. If you do short runs you will get mayo around the oil cap.
Bought an exact same car myself with alleged head gasket gone, would need a huge water container in order to drive it, having followed this site and one on the other side. I was able to convince myself it was not head gasket. it is still under test? nearly 12 months later never added a drop of oil to it.
I am about to take it off the road for belt and water pump change, complete suspension and brake change.
Only thing done to date is the tracking and camber front & rear. Very well worth doing or having done, from what you say regarding the water pump you can forget the oil cooler better to spend your money on tracking and camber you will get a very good return from it.