Car will not start

I may be wrong with this, but I suspect going from park to neutral and pushing it was not related. It sounds like you have a starter solenoid issue only if when you are trying to start the engine it does not crank. Meaning when it is not starting does the engine spin over and not fire or does it not respond at all?

Here is a guide that talks about this:

https://www.2carpros.com/articles/car-cranks-but-wont-start

As for the shutting off, that may be a different issue unless the engine cranks over but does not start. If that is the case, then we need to run through this guide and start checking fuel and spark:

https://www.2carpros.com/articles/engine-stalls

This could be a crank sensor issue or even a failing fuel pump.

So, let's start with checking the fuel pressure when this happens.

https://www.2carpros.com/articles/how-to-check-fuel-system-pressure-and-regulator

https://www.2carpros.com/articles/symptoms-of-a-bad-crankshaft-sensor

Let me know about this info and we can go from there. Thanks