Sounds exactly like a vacuum problem to me, especially when you consider the rough idle problems in addition to the brake booster problem. My guess is that you have a vacuum leak somewhere at, in, or around the booster. Check the hose coming from the booster, check the connection where the hose goes into the booster, and follow the hose toward the manifold.
Concerning the "hose that is connected under the battery (....connected to a nipple on something black) but the other end is free." This sounds like a vent hose, that doesn't really connect to anything.
