To be deep enough in atmosphere for residents to breath, while maintaining centre of gravity at geostationary (both seeming convenient off hand, although the geostationary less important - who cares if it zips through the air creating a continuous sonic boom?) the distance from centre of mass to nearest Earth approach would be thirty kilometres - for perhaps three or four layers of residence. And even then, atmospheric pressure is about Everest peak - need canned oxygen on hand. If you're going to build that far (with so much space that could be pressurised, I suppose) why not do the extra few kilometres and make it an orbital tower? Tensile strengths of the materials involved would be equivalent, and at least, we'd know its postal address, and it wouldn't be wandering over the equatorial region of the planet.
Or put it in a lower orbit, with several towers, and rotate it round its asteroid, so just the bits of the 'arms' dipping into the atmosphere are traveling at the same speed as the air. Hooks on the ends collect airships and drag them up into orbit. Obviously the arms need pressurising and there's no parachute jumping, and you might not even be on an equatorial orbit, so what country you are over at any given time is - variable. Very variable.
But there is no time scale on this. Collecting a planetoid, putting it into Earth orbit and then stabilising the orbit will take many years, probably decades, and construction of the tower can't start until a fair percentage of that stabilisation is done - moving in will probably need to wait a century. You don't want to rush things if a minor miscalculation can drop a dinosaur killer somewhere on Earth, with more special effects than a disaster movie (and no retakes). I wouldn't like to be the insurance company guaranteeing payment for the damage it could do (assuming there are any survivors to claim their payments).
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