Since your engine fails to start hot or cold, the fuel pressure is where to start. If you are waiting on the fuel pressure gauge adapter, testing the coolant temp sensor is worth a try. I don’t think you’d even have to remove the sensor. All you’d have to do is measure the resistance when the car has sat overnight, then compare the measurement to what it should be at that temperature. Make sure you are testing the correct coolant temp sensor. Many cars have two, one is only used for the dashboard gauge. It’s the one used by the computer that you should test.
Learn mode is what happens when some component is changed or the battery is removed. When this condition is detected, the ECM often goes back to some basic parameter list to use, parameters that will assure the car will start and at least run, but maybe not perfectly at first, until the correct parameters are re-learned for the new configuration.
Why? As a car ages stuff like the throttle body gunks up, various passages don’t pass fuel and air quite as freely over time, so to keep the fuel air mixture and idle rpms to spec the ECM learns and compensates. It might have to boost the amount of gas, or decrease the amount of gas compared to when the car was new. Or increase the amount of air or decrease the amount of air.
Learn mode is a good thing. It works great as long as nothing changes. But it can cause the engine to run – esp idle – poorly immediately after changing something, or just disconnecting the battery.
In an ideal world even in learn mode, the car should still start. But alas this is not an ideal world. There’s talk here of some cars, if you disconnect the battery which resets the computer to learn mode, then the car won’t start, and has to be taken in to the dealer for reprogramming. I’ve never personally experienced that, but something like that could happen I suppose.