1996 Chevy Impala My Electrical System is jacked up totally

Electrical problem
1996 Chevy Impala V8 Front Wheel Drive Automatic 120XXX miles

After installing a new "after market" car alarm last year I noticed strange problems with my car. For one the anti-theft deter wouldn't allow my car to start. Next thing, all my interior lights would stay on, so I took out all the bulbs (broke a few in the process) in order to keep my battery from dying out on me. So here I am almost a year later, and I'm still having the same problems. After installing a new alternator, starter, and battery (which has now gone bad maybe due to all my electrical malfunctions) I'm still having some crazy unexplainable electrical problems. For one, my mileage changes drastically at any given time when I start my car. But if I switch it back off it will go back down to where it previously was. Next thing, I can hear a "clicking" sound coming from down by my fuse box by my left knee. And then my dome light will continously stay on if I leave the bulb in. CAR ON OR OFF. If i'm driving or not. With all the doors closed, it will not turn off at all! I don't know what type of probably this is. I got a brand new alternator and someone said my problem will be fixed, NOPE! I got my alarm disabled, still nothing changes. If anyone has some insight on this type of problem please help me out. Thanks in advance.