If the car starts and drives EVERYTIME i would say the alternator isnt the problem, it takes a LOT more to start the car than it does to arm your alarm, most alarms these days boast the fact that the total system when armed has a lower current drain than the warning LED, (which is like less than 20mA)

Although the alternator only giving out 12.5vwhich is a little low i dont think this is your problem for the alarm, if the alternator wasnt charging sufficiently you would reach a point where there isnt enough juice to physically start the car, if u get me?

How did u test the alternator?

And also does your battery warning light work?