If I were to want to take intercept / control of what the thermostat was currently doing I wouldn't touch the thermostat wiring at the thermostat.
I would use an AC input optocoupler back at the HVAC unit for each of the controls Fan, Heat & Compressor. Disconnecting them from the HVAC unit.
GPIO input the output of the opto's, perform what ever calculations needed using the Arduino and control the Fan, Heat & Compressor using relays. I would not use the cheap "for Arduino" relay boards on my HVAC system, I'd use something more reliable. But that is just me.
Note: There will be no need for the "C" terminal at the Thermostat.