To answer the first question that I forgot about, you will not cause damage to anything by driving with a defective neutral safety switch. Mazda calls this the "transmission range switch", but it's the same as a neutral safety switch. Yours has two switches built into one assembly. One switch turns on the backup lights in "reverse", and the other one prevents the starter from engaging when the transmission is in gear.
It sounds though like you have a mechanical issue with the cable. That may just be preventing the switch from reaching the "park" position when you shift to "park". Replacing the switch won't fix that.
If the neutral safety switch were to fail completely, you can bypass it to get the engine started, but be sure the transmission is in "park". You'll look funny chasing after it if it starts in gear, then takes off without you! There is no starter relay which would make the job a little easier. Instead, follow the large positive battery cable down to the starter. Next to the large terminal that cable is attached to will be a much smaller wire and terminal. Touch those two together to make the starter engage. The ignition switch will have to be in the "run" position for the engine to start.
Thursday, April 2nd, 2015 AT 6:37 PM