In connection to this, I am using my Omega with an Arduino shield (primarily because I have quite a bit invested in Arduino shields).
Among the shields I have is the Adafruit PowerBost 500 shield (https://www.adafruit.com/products/2078) with a 2000mAh lithium ion battery.
When plugged in to the Omega Arduino shield, this powers the Omega fine.
In terms of controlling our Omegas remotely, well they're still working on getting us the apps and libraries to make that kind of thing easier. In the meantime, if all you've got is the expansion dock, you could start playing around with GPIO? I'm new to this and just having fun with that so far. Here's a good intro if you're new: http://www.instructables.com/id/Blinking-a-LED-with-Onion-Omega/
Hahaha. Wow. I wayyy over-engineered the problem. I guess I didn't have the omega snapped down all the way... I just pushed it down into the doc further and it works now. Idk how I managed to get this far (set up config etc.) with it partially unplugged enough to cause this issue. Anyways. Nothing to see here!