: Just far enough away so that they don't collapse into each other??

Sort of, one of the interesting phenomena LISA will hopefully detect is related to this. Two black hole could orbit a common center. For some time the orbit would be fairly stable, but they will radiate gravitational waves and the orbit decays. Eventually they begin spiraling into each other increasingly fast until they form a single hole. Due to the energy radiated away in the waves, the new black hole's mass will not be a linear sum of the old hole's masses. It will be a Pythagorean addition.

: What would occur at the point where the forces's of each where in exact balance??

There would be some frame dragging there. An object at the center that is not rotating in our remote coordinate frame will experience some centrifugal force (not fictitious in GR) there. This will be due to the rotation, not merely the tug of war the two holes will play on the object.