For your first concern, that sounds like a transmitter issue. Even if they are 2-way remotes, if the LED is not activating, the problem is the remote.
For the second concern, it sounds like either the relay is sticking or the relay is being commanded to stay engaged. What is the timeout setting for the start system? This is the setting that dictates how long the car will run unless you start the car. Often this timeout is overridden by the door opening, the hood opening, the brake pedal being depressed, or the key being turned to the Run position. I'd suspect that the some settings are not set properly (active high versus active low, bypassed, etc.) or you have a faulty relay.
-Rod