I have recently built a similar set-up using the RPi to switch on and off my central heating system. It was a fun project to research and design the hardware and software elements.
I soldered some fine wires onto the contacts of the relevant PCB button in the control unit, with a connector on the end so that I can disconnect the system when not in use. In my system, the buttons are actual components soldered onto the circuit board, which made it easier for me to add the wires. If they were the type that are just bare pads on the board with a conductive patch on the back of the rubber button, it probably would have been more fiddly to solder.
The wires are shorted together by a reed relay for a few seconds, which simulates pressing the button. The reed relay is driven by one of the RPi GPIO pins, via a transistor. I also added a DS18B20 single wire bus temperature sensor. All the components went on a small piece of veroboard, which mounted directly to the RPi using a 26-pin header socket to attach to the GPIO pins.
The system works well, but I did have to move the temperature sensor off the board on a long lead. My RPi runs at about 50 dec C, and there wasn't sufficient airflow to avoid the CPU warming up the sensor.
I didn't try adding a webcam, but I don't see a reason why it shouldn't work.
I found various guides on the web helpful for driving the relay, monitoring temperature and setting up a secure webserver to control it all.
For fun, I also set up a
cron job to record the temperature every 15 minutes, and a web page that used the Google Charts API to visualise the result.