[aerohead]

Does it have an automatic trans?
A friends Corolla developed 'strawberry milkshake' transmission fluid.
The transmission cooler inside the radiator had breached, allowing the two fluids to comingle.
The engine oil was not affected.
Transmission clutches would not 'grab.'
After a $220 radiator and continuously flushing the trans with new fluid as the nasty stuff pumped out of the coolant line, she cleared, and transmission service was restored.
If transmission fluid got into the cooling system it will cause an overheat.
Heat cannot easily pass through a coating of oil, into the coolant, preventing the heat-rejection.
Ethylene glycol coolant is petroleum based.
If it's not changed at the proper interval, it bio-degrades back to 'oil'. According to our old auto shop instructor.
With improper coolant, the electric -fan thermo-sensor, ECM: ignition timing and electronic fuel injection , would be processing bogus voltage signals.
It might explain the rough engine and overheat.