I would recommend getting an Arduino Mega and a Smart Things Shield (Assuming they ever come back in stock… if they don’t, this will be much more complicated…)
If you can get a Smart Things Shield, you can attach it to the Arduino. Then, you can take the wires from your ancient alarm system, ti all of the “ground” wires together and attach them to ground on the Arduino and then connect the “positive” wires to individual pins on the Arduino.
Then, use @ogiewon 's garage door opener project as a base.
You can remove the garage door buttons and sensors from the code and just use the door sensors. You can expand and rename them to match your house.