There is a plastic switch mounted on the steering column that tells the ecm when the car is in park and drive. It also turns on the reverse lights. Altho it can go bad, it is activated by the steering column which provides a mechanical lock out for the starter so that it will only crank in park and neutral when properly adjusted and which locks the column when the car is in park and the key is removed.
From the sounds of it, your first problem is with the adjustment linkage from the tranny to the column.
Get under the car and find the rod that runs from the tranny linkage to the steering column "ear" just in front of the firewall.
Loosen the clamping bolt that holds this rod in the tranny end so it can be adjusted.
Get in the car, Make sure the shifter is fully in Park. Then reach down on the column behind the steering wheel and grab the "hub" that the column shifter would be attatched to if it had a column shifter. Rotate the hub fully counter clockwise which should lock the column and allow the key to be turned to off and removed.
Get back under the car, push the linkage rod that goes to the column toward the firewall and tighten the clamp bolt while holding it.
In theory, this should allow the key to be turned to Off while in park and be removed. At the same time, this should turn on the reverse lights.
Pull the shifter into reverse and see if your reverse lights come on....I don't remember if the key has to be turned on for this to happen, or not. If they don't come on, try moving the shifter forward and backward to see if the lights will come on when in a position not quite in reverse, etc.
