Saturn LW200 2001 will start &run then die.

If the position sensor for transmission is faulty it cannot detect park or neutral. In my younger days we called them neutral safety switches. These days the key locks in ignition too when they die or come loose.

My 2003 Saturn l200 2.2 ecotec did this when the battery was dead. Put some juice on the battery, then the car came out of park, and I was able to remove the key. To the car, there's a vast difference between a sluggish battery and a dead battery. It may have enough amps to crank it, but it wont start w/out full output from the battery.

You are on the right track by finding the injector pulse missing.
The computer grounds the injectors based on signals from the cam and crank sensors. You may have to use a scanner to check for trouble codes in the computer and watch the sensors work.
The problem could also be a relay or the key switch.

check the inter lock system the key must be all the way off the linkage could be out of ajustment.

If the shifter won't move, you could have a problem with the brake switch or the solenoid that unlocks the shifter.

Broken shift cable ?
If the neutral safety switch is on the transaxle and the cable is broken, the switch could still be in the reverse position.

For the original question, we had the same issue with a Saturn L100 - the fix was installing a new BCM - (Body Control Module).

these have a safety system so the shifter won`t go into any gear unless the brake is pushed(indicating someone sitting in the drivers seat),if the brake lights aren`t working this indicates that the fuse that operates the shift lock solenoid and brake lights is not working and needs replacing.check the fuses and replace the fuse and things will probably return to normal.

Almost all newer cars have a safety that won't allow you to shift without a foot on the brake. These are either cable or electric solenoid activated. They fail, so check your owners manual, as it will have the procedure to get it to shift, untill you can get it fixed correctly.

check the alternator and the battery...