Cranks but will not start

Two-digit fault codes are used on 1995 and older models. You should be getting three-digit fault codes.

Thank you for including lots of detail, especially that this is a single overhead cam engine. There is a somewhat common problem that I ran into once myself, and the clue was later found to be the code 37. That has absolutely nothing to do with the problem, and I do not know why it sets that code, but it does. On the 1996 and newer engines, you will eventually trigger a code "cam and crank sync", which implies the timing belt has jumped a tooth. In fact, if the timing belt jumps one tooth, the Engine Computer will detect it, turn on the Check Engine light, and set that code for cam and crank sync. At two teeth off, it shuts the engine down to protect the valves because this is an interference engine. If it were to make it to three teeth off, valves will be bent.

Start by checking the timing belt, and when the marks look like they are lined up perfectly, the common cause of setting the cam and crank sync code is there is a dowel pin between the camshaft and its sprocket. That pin shears off, then the sprocket can gradually slip and rotate on the camshaft. While the timing marks are correct, that only means the sprocket is in proper time. The camshaft is slightly retarded, and the camshaft position sensor is on the opposite side, on the driver's side of the cylinder head. The Engine Computer just knows the camshaft is late. It does not know why. If you are right between the equivalent of one or two teeth off, you can have an intermittent problem.

As for the fuel pressure, you cannot really go by that for clues. The computer runs the fuel pump for two seconds every time you turn on the ignition switch. That is why you have fuel pressure when you have the crank/no-start. It turns it on again during engine rotation, (cranking or running), and it knows that by the signal pulses it receives from the camshaft position sensor and the crankshaft position sensor. It is also the mismatched timing of those pulses that tells it when to shut the engine down to protect it. It does that by turning off the automatic shutdown, (ASD) relay. That relay powers up the ignition coil pack, fuel pump or pump relay, alternator field, oxygen sensor heaters, and other stuff. If the computer decides to shut the engine down while you are driving, that fuel pressure might be used up and drop real low. Pressure could be low when you check it one time, and it could be normal from turning the ignition switch back on at a different time. Also, just watching for the spray from the test port never gives reliable results. I was told I was being tricked many times by people smarter than me, and I found them to be right. You can get a huge spray from ten pounds of pressure, which is way too low for the engine to run on, and at times 40 pounds of pressure can look like it is too low.

As I mentioned, you should be getting three-digit fault codes. If you need a list of the definitions, go here:

https://www.2carpros.com/trouble_codes/obd2

If you really are getting two-digit fault codes, here is the list of definitions specific to Chrysler models, 1995 and older:

https://www.2carpros.com/articles/retrieve-trouble-codes-for-chrysler-dodge-plymouth-odb1-1995-and-earlier-car-mini-van-and-light-trucks

1995 was the first model year for the Neon, but they actually were on the road already in early 1994.

You owe me a cookie, ... Chocolate chip, please. Not for the answer but for being so darn cute!

I do not know why you are getting two-digit codes. They should be showing up in the odometer display. Someone must have replaced the driver's door with one from a 1995 model. That means the door sticker will show a manufacture date of 1994, and therefore it is now a 1995 model. Best explanation I can come up with.

To add a couple of points of great value, to get a fault code related to the cam or crank sensors, it is usually necessary to have a stalled engine that is coasting to a stop while you are driving. That gives the Engine Computer time to detect the missing signal while the other one is present. It seems almost all the time, if you erased that code previously, it will not set again just from cranking the engine. When that happens, we connect a scanner to view live data and see what the computer is seeing. I have a Chrysler DRB3 for all of my vehicles. That lists each sensor with a "No" or "Present" during cranking to show if the signals are showing up. I am pretty sure aftermarket scanners do something similar.

The last exciting note is if you recently disconnected the battery during any of the work you did, idle speed will be too low. The engine may not start and run unless you hold the accelerator pedal down 1/4". You will not get the nice idle flare-up to 1,500 rpm at start-up, and it will tend to stall at stop signs. "Minimum throttle" has to be relearned before the Engine Computer will know when it must be in control of idle speed. That is the only thing that does not get relearned right away automatically. When this happens, to meet the conditions for the relearn to take place, drive at highway speed with the engine warmed up, then coast for at least seven seconds without touching the pedals.