The car starts if you provide twelve volts to the solenoid while the igniton is in the run position. The wire that normally powers the solenoid does not show any voltage when turning the ignition to start. Its this conjunction of lack of fuel and lack of crank that has me looking at the BCM/PCN communication as the probable issue as this is exactly what it does when the security mode is triggered.
I've attached a drawing for the starting circuit. To be clear, you are getting the no-crank condition, but the SECURITY light is NOT flashing, correct?
If that's the case, I would trace out the starting circuit. Make sure you see the CRANK signal go high at the PCM when the key is turned. Verify the fuse in that path is good. Verify the signal between PCM and starter relay in the underhood fuse box. Verify continuity between that relay and the starter solenoid. In short, you need to figure out where the break down is in that path.
Once you get it cranking reliably, then shift focus to idling/dying.
When you do continuity checks (ie, measuring ohms between the ends of a wire) I recommend having the battery disconnected.