I haven't found an onvif event for doorbell press yet, but I didn't look very hard.
Alternatively, to send doorbell events, you can install a relay parallel to the power wires. When the doorbell button is pressed, it will cause a dead short on the power wires. The voltage drop will trigger the relay. You may have to change the chime type to "electronic" or "mechanical" in the app. If you use "no chime" the button will not change the voltage.
From there, the relay can be wired to a simple switch to connect and trigger whatever home automation sensor you may have (i.e. alarm zone, zwave door/window sensor, ZigBee, etc)
P.S. the PIR doesn't appear to actually work. This was reported by other people in this massive thread. You can verify by covering the lens and moving in front of it. The camera won't record motion. It seems to be using software motion detection.