well if you installed (vs had installed by some one else) a stereo then the first thing i would do is Un-Install The stereo. One wire at a time.
Lets think.... (Thump/thump/thump...) Ok, Dome lite comes on when either a door switch closes (Doors open) and a contact is made, Or when you turn it on via a inside dimmer /lite switch. If neither of these are happening then I strongly suggest that you somehow tapped into a wire with your stereo that you shouldn't have. Maybe the dome lite. Power is feeding back into that circuit making the car think a doors open. The remote starter wont start the car remotely unless the doors are locked and car in park. Hence it thinks doors open, don't start (Safety) wha....la !!!
easy as pie. disconnect stereo 1 wire at a time and you'll find that all of a sudden dome lites out and starter works. then hook Stereo up correctly later.
Hope this helps... Rock Base