Yes the Bose sub runs off the same amp as the rest of the speakers, and yes unplugging it will help the other speakers sound better.
You aren't going to ground your sub to the electrical system are you? You need to put both of your speaker wires into your amp output, not the electrical system.
Also, if you meant somewhere to ground your sub amp, I would suggest that you ground to the same place you ground your cap(s). This will prevent ground loops. It is generally agreed upon in the car audio sector that the best practice is to use a single ground point.
I would also like to mention that caps are pretty pointless unless you have upgraded everything else first, and are looking to spend more money. I have a system that can suck about 1000W, but I don't have a cap, and unless I bump at full volume for extended periods while at idle, I have no power problems whatsoever. If you want to know more about what I mean by everything else, let me know.