Well we can waste your time and tell you to screw around with things that wont fix your problem or you can listen to everyone when they tell you its the syncros. I had the same problem and im sorry to tell ya its not always an easy cheap fix. Its just the way the DSMs roll.
I personally don't know how to test an alternator if it isn't hooked up to a running engine. If it starts smoking once the battery is hooked up to it, I think its safe to say you need a new one.
what? why would you waste your time? both of these projects seem like a waste of time.. if your gunna turbo it.. turbo it but since you have a GS and they arnt that expensive, get one with a manual and if you want AWD get a GSX. Just my opinion.