[soundman98]

and a short press of the t/c button will illuminate the light, but then turn off at 25 mph i believe.


of note, none of the issues listed have anything to do with any systems or wiring that any remote starting system would connect to in any regard. remote starters connect to 7 primary circuits in the vehicle. starter, power, ignition 1, ignition 2, parking lights, brake pedal, and immobilizer.
starter obviously starts the car.
ignition 1 and 2, sometimes also referred to as 'accessory 1 and 2' are triggered circuits that control power to things like the radio, hvac, and window controls, among other things. these circuits usually turn on with the key, so the remote unit has to imitate the key being there to make the engine ecu work.
parking lights are used as a visual indicator of alarm functions.
brake pedal wiring is used as a failsafe/disable to the remote start function.
the immobilizer circuit is usually entirely bypassed via a few different methods(all keys now have a digital chip that communicates with the car to verify that the key being used is legitimate, without this digital communication, the car doesn't start), this fakes the car out into thinking the key is present to allow the starter circuit to do it's thing.

as i said, none of those connections would only effect the issues you list. if any of these things were improperly connected, the number of things wrong would be much more significant than 3 items.