Is there a way to quickly move the Triball to the center of a negative shape (block hole, for example, especially if only a part of it intersects a positive shape)?

Or when the negative shape does not intersect a positive shape at all, and is just floating in space, ready to be positioned?

Yes,
in 2012, Hold the Shift key and drag the center handle, plane to a cylindrical face or edge.
tom

attached is video
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