If my car wont stay running what could be the cause?

I would check for fuel pressure. I would about bet the pump is on the way out, that is what caused the relay to melt. Pump is probably pulling too many amps. Anyway, I suggest checking fuel pressure at the rail and if you have the capability, to check the amp draw of the pump.

Vacuum or incorrect air fuel ratio .also if you moved the butterfly inside the throttle body you will need a throttle position sensor .

I’m happy to assist further over the phone at https://www.6ya.com/expert/joseph_cfbe9ad29efdc8bb

Try TPS. Throttle positioning Sensor. This sets base idle. (Unless this model is too old for that sensor)

If it feels like a stick shift car when you come to a stop and don't push in on the clutch and the engine lugs down and dies, it's most likely caused by the torque converter not unlocking. There is a solenoid that causes the converter to unlock and lock. That is the most likely cause. It may, also, be a broken wire or bad connection.

glow plug fault. check the relay. then the glow plug.

Can you hold throttle pedal down a bit and stay running? EGR might be stuck open. Vacuum leak from broken hose will cause this.

If your vehicle is fuel injected I would say you have a blown fuel regulator. This would cause excess fuel into the cylinders (flooding) causing a miss and coughing and also blow fuel into the air breather. If will also cause the vehicle to run poorly and could cause the vehicle to stall. Also to much fuel on startup can flood the cylinders and cause the vehicle to no start.

idle speed control not working. if idle speed too low engine wont stay running.

Here is a link for you to retrieve your DTC codes on your Lumina.
http://www.troublecodes.net/GM/

Your Catalytic Converter is Clogged up. This is what is Causing it to not run.