It is normal for the oil pressure light to stay on with the key in the on position. If the oil pressure is causing the engine to shut down, there should be a code in the ECM for low oil pressure. Also, if the engine will not start back up after it shuts down, a low oil pressure warning will not cause the engine not to start.
It sounds like you have either a power supply problem or faulty ECM. I have seen this before.
When an engine shuts down and has no check engine lights come on and will not restart, most often I find the ECM has lost power or the ECM is faulty.
Please reply back if you need more help or have more questions.