re: "...If so, then start the engine and disconnect the battery and then...."
Oh, no! Don't do that! Don't even think about that -- ever!
If you ever have an engine running and suddenly disconnect the battery, the spike will fry virtually every electronic system (radio, fuel injection processor, etc.) in your car -- instantly -- as well as destroy your alternator's diodes, so if your alternator wasn't bad before this test, it will sure to be toast after it.
You may be able to get away without damage by doing this on a lawn mower, but NEVER on a car.
I sure hope that I caught you with this warning before you tried this.