I'm kind f out of my element with the Grand, most of my experience is with the XJ. But some (likely much) of the system is basically the same with minor differences (OBD I and OBD II).
I've had my alarm system shut down the ASD relay before and even had it partially work. The ASD function was a sometimes thing (it didn't work, it did work and it kind of worked but very poorly, engine ran like crap).
But I also had a short in the CPS (which was actually on the non test side of the CPS and the ground, which wouldn't show up on a normal CPS ohm test, drove me nuts). I finally caught on when I did a simple volt test in the 5 volt supply circuit (for various sensors) at the sync sensor, noticed low voltage and then traced it to the CPS (but it likely could have been a short anyplace in the system causing the same symptoms).
Since you mentioned doing the thermostat, the first place I'd look is the temperature sensor. Inspect the wiring, do an ohm test. Then I'd look at any harness or wires in the area. I use a flashlight, even in daylight, it helps focus my vision and aids in inspection. Not sure about the grand, but on the XJ the hrness that goes down the front of the block is a known trouble spot, it gets real close to the exhaust manifold. A little nudge can can cause it to move enough to cook.
IATony gave a good piece of advice, *grounds*, simple but important. A ground wire gets cut, makes poor contact or the nut is loose. I've had a lot of grond issues in my XJ's, likely the same in the Grand. Check the block ground and the engine to firewall and/or front clip ground (at the very least). A good ground diagram would be helpfull, there are likely a dozen grounds, all together in the engine bay, near the dash or on the firewall (all of them important).
I have both a plethora of PDF material I've collected over the years, Alldata and FSM's on my XJ's. You really do need some sort of reference material to troubleshoot effectively.
I've found shaking a harness will sometimes show up ground faults, on grounds that appear good and even test good with an ohm meter.
I'd hate to send you down the wrong path and waste time, but all I can really do is help you play the percentages some and share some experiences.
I'm pretty sure the check engine light should come on (bulb check) when the key is turned to the run position. If you have the digital dash, chances are you can do a trouble code check, I'm not exactly sure of the method for the Grand. If you have the analog dash, not much chance of getting any codes (especially since your check engine light isn't working and your likely OBD II anyway). Many of the chain parts stores have code readers (though they can be far from definitive), that may steer you in a fruitful direction.
