HELP! 1987 Turbo

[Toyota drive]

I just baught 87 turbo, 5 speed Supra that has some minor problems. The first night i had her, the factory alarm went off. So i got in and turned the key and it shut off. Then when i went to start it, it didnt do any thing. So after a while of trying to figure out what to do, a friend and i bump started it. She ran fine after that, but i still had to bump start after shutting it off. So i went to a friends house and disconected the negitive battery cable for a few minuits, put it back on and she started right up. It ran good the rest of the night. The next night, i was driving down the road and went to make a turn, so i pushed in the clutch and it started to sputter. I stoped and it started to run really rough and then died. I tried to start it again and the starter did nothing. All of my accessories worked but not well at all. So after a bit of playing around i replaced the battery and she started right up. I drove it for a day and now she started to run choppy again, and it shut off. It wouldent start again after that. I hooked up a battery charger and she started fine, but only ran good for a few minuits, until she finally died again. Im guessing that i have a bad alternator. But also i have a fuseable link where the alternator fuse is in the fuse block under the hood. Im really hoping for some suggestions and also i need to know how to disconnect the factory alarm system. Thank you very much, Danny.