Hi
Very hard without plugging in myself. The alarm should turn off even with a fob that's not working as soon as you turn on the ignition with an authorized key.
It's not going to be the alarm module as the car should still start and run.
SKIM/ immobilizer fault the car should start and run for 3 seconds or so then stop but then after that may just crank.
Alarm is not part of the PCM it's under the top cover of the dash. Immobilizer is in two parts a receiver SKIM module around the barrel and must send a signal over the bus to the PCM to allow it to start.
If you have a coms failure you would get all the fault you are describing, is there any history to this vehicle? did it just not start one day?
The fact that the SKIM light on the dash is on (little red light) points to a SKIM fault, PCM fault (when I say fault could be the unit or wiring to it) or communication between the two.
Sorry buddy but I think without being able to get any proper fault codes and diagnostic done I can't point you in any direction.

 

I have a very similar issue with an 01 T&C. I'm going to see if using a different key helps, i have the solid red light on. Is there a way to pull the codes myself, or do I need a special scanner? I'm on a very tight budget and really don't want to go to the dealer.