From the additional information it sounds like a bad connection in the starting circuit. The alarm seems to be operating correctly and it is detecting the key as the light doesn't stay on or flash when you attempt to start it. The key light means there is a problem in the starting circuit as well. Being an automatic it doesn't use a starter relay other than the one under the dash that is part of the anti-theft. If it has a bad contact it could still detect a good key and disarm but it may not start.
Below I attached the starting circuit and relay box. E29 is the horn relay.
The antitheft is the last image.
Images (Click to make bigger)
Saturday, October 13th, 2018 AT 10:50 PM