^take a look at the 10th pic in Op's post. You can see the tube that goes into a black elbow on the positive side of the battery. The reason why they left it hanging is probably the battery they installed has the vent located in the center of the battery and couldn't reach. Take it back and tell them to hook up the vent hose. It is not there for show.
If no codes are being thrown use Chevron Techron fuel injector cleaner (concentrate). It solves rpm fluctuating upon cold start-up. Also, for most BMW problems start off by scanning your car with the Peake Research Tool. It contains the actual BMW codes. If you want to register a newly installed battery for free (just buy a $10 cable) and google/download BMWLogger