Okay. Your #3 makes this much easier! (Kind of).
Yes. You use the Arduino to read the state of the door sensors and @ogiewon’s library and sample code to communicate with the thing shield, which communicates with ST.
For the door sensors, there should be two wires for each one attached to the alarm panel. Pull those three pairs of wires out of the panel and run one side of each one to a digital pin on the Arduino Mega. Tie the others together and then run a pigtail from there to a ground pin on the Arduino.
The ST Anything libraries will monitor the state of those digital pins and when they change, ST will know it and you can use that to trigger things like any other door sensor.
The motion detectors are a little harder. They need power to operate and the Arduino probably cannot provide that power. So you will need to use an additional power supply. if the motion sensor has four wires, then there are probably 2 for power and 2 that would get wired in just like the door sensors. If it only has 3 wires, it gets even more complicated.
I am using this LCD display:
It takes 4 wires and I used code here to learn how to write to it:
The ST Anything libraries don’t handle the display. I added my own code for that. I actually had to replicate some of the sensor tracking code that the libraries handle so I could tell when to update the LCD. It’s not ready for public consumption yet. I’ll try to clean it up over the weekend and post it on github.
But, really, that should be the last thing you would want. I’d focus on getting the other stuff working first.