The thing to get is a multi-meter. Anywhere from $20 RadioShack to $200 Fluke. But then you'll have to learn how to use it.

You say you're a fair mechanic, so it can't be too hard. You just have to verify that any particular wire actually goes where it's supposed to go, isn't disconnected somewhere in the middle, and it doesn't touch any other wire or the chassis.

You'll also want the wiring diagrams for your car. Helm manual...

The engine management shouldn't have anything to do with the dome lights or the door-open warnings. Maybe those problems are unrelated, but it just means more than one electrical thing is connected where it shouldn't be.