2005 Saturn Ion Serious electrical issues

I am having trouble with my Saturn. Last week, I was in my house and notice my car alarm going off, it scared me a bit since this never happens. My bf came home 2 hours later and my car alarm went off again and this time my dome light would not go off. Ten minutes later my alarm went off again. So we decided just not to lock the car because it was midnight out and I am sure my neighbors were tired of hearing my car alarm. As the weekend went on, I notice many things not normal happening with my electrical system: when I have the keys in the ignition and the doors are shut the dinger continues ding, my dome light stays on all the time when I am not in the car but will not stay on when I open the door, and I cannot manually turn my dome light on and off.