OK. Jump the S post of the solenoid and check that the starter turns. If that works, then check voltage at the signal line when trying to start. If there is no voltage, then either a break in the line, bad ignition switch, or the ECU is commanding a no start condition, which would be odd with the fuel pump energized.
If the fuel pressure won't stay, then there is a leak in the system. You haven't said anything about fuel odor, so I assume the lines aren't leaking. So check for a leaky injector (if found, change the oil before running), the ball stop, and the sensor/switch (part of the pump?) that determines that pressure has been reached.
There have been lots of odd-ball issues over the years, so I'm very curious about what you find.