The ATF in the cooling system is minor and it won't harm any gaskets. You should have flushed the cooling system when the radiator was replaced, to remove the residual ATF, but it's not a terrible thing.
The oil pressure gauge is an ON/OFF device. It reads zero when the oil pressure is below a dangerous level such as 5-10psi. So the pressure sensor in the side of the engine is only producing a signal to the gauge when the pressure is adequate, above 10psi etc. If your oil gauge reading is right, the pressure is dropping dangerously low, often.
It would help to confrim the pressure is low, with a separate gauge and sensor, or by replacing the OEM sensor, and maybe the gauge if that is also suspect. But be prepared to need to pull the engine back out, the pump or the pickup may be not right, or some other internal issues(too large bearing clearances etc).