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I'll try to answer your questions, let's see if it helps!
the Omega is a very small computer, but just like your desktop/laptop it has a wi-fi connection. This means it can communicate (and others can communicate with it) through a local network (your home wi-fi) or even through the internet. One difference is that you can pair the Omega with the expansion dock and then also use it to interact with hardware (temperature sensors, infrared sensors, you name it!).
Yes, there are numerous ways you could approach this. You could have a mobile app on your phone with a geofence setup around your house, then everytime you enter the geofence the mobile app could communicate with the Omega via 3G/4G and the Omega would turn on the AC. Another alternative is using the Omega with the Bluetooth (BLE) expansion, this way the Omega could detect the presence of your smartphone and also turn on the AC. One important point is that your AC needs to provide some way to be turned on, easiest one would of course be via wi-fi. Other than that you would probably need an infrared blaster connected to the Omega so it can simulate the AC remote control.
I don't know how Zwave and Zigbee work but you could probably have some kind of antenna (or sensor) connected to the Onion that can pickup their signal and then you could do whatever you want with it from the Omega.
I don't know how much experience you have with this kind of stuff but one thing to have in mind is that none of this will be super easy . It will demand some knowledge of electronics and software development, but the internet is full of tutorials and examples!
