Angela,
You mentioned in your original post that the car has a factory alarm. I picked up my 951 for a song because it had similar problems to your daughter's car. Well, after chasing down through the wiring schematics, the factory alarm became my main suspect. After disabling the alarm, the car started right up. Your problem may be similar.
I looked on the MX6 forum and found that disabling the alarm is an relatively easy task. You might want to give this a shot and see if that addresses the problem.
Disable factory alarm--easy way! - Mazda MX6 Forums: MX6 Forum