In this situation it is not a good idea to just keep throwing parts at a problem and hope it gets better. Its an even worse idea to trust what the workers at Autozone/Advance/Napa tell you. They are not mechanics, they are cashiers and I've had some... experiences to say the least and now I just go there armed with part numbers so there is no debate.
At any rate, you are probably going to want to take the car to a Ford dealership and let them do a diagnostic at least. That will at least pinpoint what is wrong and then if you feel it is something you or your son or a friend can do then you can buy the parts yourself and do it, or just have Ford do it. There is a very good chance the alternator was just shot to begin with and its causing all these other issues.
I hope you get it all figured out, you can remove the alternator which is very easy on a 2004 and take the unit in to the parts store to have it bench checked. That right away will tell you whether it is bad or not. If you still have the receipt somewhere then they should warranty it. Good luck with it and let us know if you get the codes pulled.
2000 Mustang GT Steeda #0048 34k miles. 99 Cobra 4V swapped, FTBR IRS swapped, exhaust, tuned by James Gordon.
Just because I give you advice, doesn't mean I know more than you. Its just means I've done more stupid ****.