Do I have a electrical system problem

I have a 95 nissan pathfinder. It has abou 210,000 km on it. My dome light inside remains on all the time and the door ajar light on the dash is always on. The rpm gauge slowly moves up until it is maxed out, but the engine does not rev. I have also had to pull the fuse on the horn and car alarm because they both would begin to sound when the car is turned off and parked. Do u have any ideas as to what could be the cause of this problem, or do I have a serious electrical problem. By the way, the car runs great.