I don't have anything like a full answer, but a couple of points: 1) your starter sounds fine to me, it sounds like the electric starter motor turns over the engine every time, just sometimes the engine won't start running, right? 2) It's normal for the oil pressure and alternator lights to be lit up if engine rpm is zero and the ignition is turned on. I'm less sure about check engine and A/T temp but probably also.
