I've had all kinds of problems on vehicles with aftermarket remote start systems installed. Usually as a result of poor connections/installation than the system itself. If you can't use it anyway, I would throw that thing right in the dumpster before anything else.
I'm almost completely positive that that car would have an immobilizer, which means that's another computer that can interrupt the starter signal. Most cheap remote start installs on immobilizer cars involve attaching a spare key inside the steering column trim near the ignition to fool the immobilizer.
Is this a manual or automatic transmission car?
Yea, pretty simple circuit, and it sounds like you have fairly predictable symptoms. Get a multi-meter and/or test light and a wiring diagram, and start isolating components of the circuits.