There is a device (probably several) controlling the fuel pump relay - the schematic should show this..Do you have the schematic of the reply itself ??
Maybe a roll-over sensor...

Chrysler strategy is to not energize the fuel pump relay unless the PCM see's crankshaft reference pulses.

Before I write a book, how good are you at electronics? I'll tell you up front "ohming" out wires isn't an indication of electronics experience. Frankly, thats a strategy that is overused, and missused. A single strand of wire making connection over a break in a wire will ohm the same as a complete wire, but there is no way it will carry the current that a circuit needs to operate.

So that being said, first question.
Does the check engine light come on when the key is turned on?
Does the check engine light go out when you are cranking the engine?
By chance is the car equipped with cruise, and have a cruise lamp, and if so is it flashing while your cranking the engine?
Are there any trouble codes?
Do you have a scan tool that shows serial data, and displays crank signal, cam signal, and sync?
Do you have a 5v reference voltage to the TPS?
Do you have an 8v reference voltage to the Crank sensor, Cam sensor, and VSS?
Do you have an ocilliscope so that you can look for the square wave created by the CAM and CRANK sensors at the PCM?
Do you have a service manual with the PCM schematic so that you can identify what pins connect to what sensors?

So, this system is far more complex than just a "roll over" sensor which Chrysler may not use or even a crank sensor.
Evidently, the fuel pump relay is energized after the engine spins over, not before (the old way), and better than using a roll-over sensor..
I have read about people complaining about paying $90 to have the "codes read", whereas AutoZone will do it for "free"..
But, in all that you have described here, John, may well be worth the $90..
Certainly the guessing method has cost a lot more than $90, and thus far only waste has resulted....

Thats the thing, "having codes read" is barely one step in the routine of performing diagnostics. This circuit really isn't too complicated for someone that is both trained in electronics, and already familliar with the circuit. For anyone else, it's probably one of the most complex "systems" they have ever encountered.

You should see what newer cars have in comparison to this one!

I saw that there is a recall for 95 neons because moisture was seeping into the PCM and causing problems. I'm wondering if that is my problem. My speedometer flies up and down when I drive in the early morning or during rain. Then one day all of the gauges dropped then went back to their appropriate positions. Then this morning, my temperature gauge suddenly excelerated through the roof and a bell went off. I pulled over, checked the engine and it wasn't even hot! I started the car and the temperature gauge went back to normal. If it's the PCM, can I just buy one from ebay?