Look at Design Pattern: Motion Sensor Timer for how to deal with how to time the alert only when the door has been open for a certain amount of time. In this case the door sensor is the motion sensor.
I can’t help with the Google Home but suspect you will need to look at the Chromecast binding. I think a GH shows up as a Chromecast.
For the iPhone app, you will need to set up myopenhab.org and the openHAB cloud connector binding. As an added bonus, you will also be able to access your OH UIs remotely safely.