Need more info! I am assuming this is an auto tranny? If so you have a rod that is connected to the bracket on the tranny that moves the arm on the tranny for gear selection which your shifter cable connects too also. The rod moves(or turns) the column sleeve when you move you shifter from park to reverse and down the line. When you shift to reverse (watch your column sleeve) you will see it moves, this neutral safty switch will also turn your reverse lights on when you shift into reverse and also keep you from starting the vehicle if it is in a drive gear and as you found out not allow you to start the vehicle unless you are in neutral or park. You can bypass this, but you will have to get a kit and install it on the shifter housing itself and reroute your wiring to there, see this Shiftworks . There is no reason for you to do this unless you are doing a tranny conversion, just replace the switch. Before you do this, check and make sure that your linkage running from the tranny to the steering column (in front of the firewall) is actually adjusted properly and not the cause for the switch not working. If you disconnect the rod, connected to the bracket at the tranny selector arm (side of transmission) and then go and twist the column (it will move both ways) in front of the dash to the extreme left (I think its left) that should be the position that is telling the safety switch that you are in park. Remove your jumpers you put on the safety switch and try and see if it will crank over. If it does crank over, your switch is good, then hook that rod back up to the tranny bracket and make adjustments to the rod above that one, that comes down from the steering column in front of the firewall. There is a nut on the bottom of that rod that holds the rod in a retainer. Loosen that nut and it will allow you to push the rod up or move it down. Push it upwards until it stops and tighten the nut back down (make sure your tranny is in park while doing all this). Now recheck to see if you can still crank it over while in park and also check to see if your reverse lights come on when you put it into reverse. If none of this works, then I would say it is your switch and I would just replace it. You can do a google search for 1969 Firebird neutral safety switch and find a lot of listings to buy one.
On your second question, I am not sure what you are referring to because you say shifter cable not allowing you to shift gears. When you unhook the cable (unhook it from where? down at the tranny or up and the gear shifter?) Am assuming you unhooked it from the gear shifter at the console. If thats the case get under the car and see if the tranny shifts at the shifter arm that the cable is connected too at the tranny (same place that rod talked about above connects to. If it does move through the gears then you can get another cable at the link I posted in this post. If the shifter arm at the tranny doesnt move, then you either have some binding with the neutral safety rods from the column or the tranny might be siezed...hopefully it isnt the tranny.