Sure, you can use battery power easily: just connect a 9 to 12V battery supply to a connector that fits the Arduino power jack. Be sure to observe the polarity: plus goes to the centre pin.
The regulator on the Arduino board will take a minimum of 9V and a maximum of over 25, but it will run a bit hot the higher you go. Don't go below 9v as things will get unstable. I would say that a 12V gel cell is your best bet, assuming you will want to run some peripherals like servos, LCD's or LEDS.
Massimo will know about the Bluetooth version, I think he is over there in Italy working on it right now, although he might be out for coffee, they have very good coffee in Italy. Either he or David C. will have the answer.