This car uses a metal key without a RFID tag so swapping keys does nothing if you get them to turn. RE the original problem still exists. ( RFID started in 97 )

The battery in the key is for the remote lock only. Even the later smart key does not require a battery to function in the ignition.

The ignition switch ( the electrical part not the key part ) could be failing and causing the no spring back. Pull the sw off the column and see if stuff turns.

I'd have to see a wiring diagram to determine if the ignition coils / injectors ECU are on the same or separate sw contact. RE if they are all on the same contact and you have inj pulse, the other are being fed from the sw.

Neutral / Park is a non issue if the car is cranking. The neutral safety sw prevents cranking only. The alarm system on this car should be a starter interrupt only but don't quote me on that.