I have not tried it with HDMI streaming. However, I know that plugging the USB cable into a computer will shut off API access to the WiFi, so I'm assuming that if the THETA is connected to a computer with USB, the THETA knows that it is connected with a USB cable to the computer and changes characteristics.
If you plug it into a wall outlet, make sure that the wall outlet adapter is capable of a 1.5amp charge. Certain rechargeable battery packs (the ones about the size of two decks of cards) will output sufficient current to charge the THETA.
I don't know if the THETA can be charged from USB with HDMI plugged in, but I think it can be since I seem to recall people streaming the THETA with HDMI for months in non-stop operation.