\rHI Ben:
The BTR-800 only works on AC power with the Power Supply plugged into AC Outlet. The BTR-150 has an optional battery pack that you can use versus the AC power supply. But there is no connection between the Battery Pack and AC Power Supply so you can use rechargeable batteries but they will not recharge while in the battery pack plugged into the bottom of the speaker. You would have to remove the Rechargeable batteries and place them in to an actual Rechargeable battery charger. Rechargeable batteries hold about a 60% charge versus the Alkaline's 100%. In regards to your second question; the Master or Sync Rock contains both the Bluetooth Receiver as well as the Stereo Power Amplifier. But only one of the two channels is connected from the Amplifier to the internal speaker of the main Rock. Not sure if you have an option to send a mono signal from your Bluetooth device.Otherwise you would only be listening to either right or left.
Good Luck
TL/3749\r