I connected the servo and a few wires to the Arduino Yun to simulate the door button and door sensor. I'm happy to report that I've now got a simple program to open the door AND a small program that can ping my phone. Running shell commands, and capturing results was surprisingly easy and straight forward. I could share code if anyone is interested.
What's next is to put them together and make an enclosure to neatly house my (eventually) complete circuit.