Yes, only 2 Cell LiPo batteries are supported. Note that the charge circuit is setup to charge at 1Amp, so make sure you use a battery that can that charge rate. If you use at least a 1000mAh battery you should be fine.
-Longer lived batteries: You can use larger capacity batteries (e.g. 1500mAh, 2000mAh, 3000mAh, etc.) as long as they are 2 cell Lipo.
-More power draw (e.g. motors): The on board motor drives can drive about 1.2 Amp per motor which is really only enough for small motors, anything more and you'll burn the drive chip. If you want to drive bigger motors you'll need external motor drivers that can handle the current.
-BBBlue via the barrel input: The barrel input is really to charge the battery or run the board without a battery. It is not intended as a way to provide more power to motors. If you need more power for motors you need external motor drivers.