Devil's advocate: not a terribly large amount of air up there, or maybe they raise it up when they're really screamin' through the upper atmosphere? And I do think the asteroid capture thing is going to happen within two or three decades, for sure. Won't be a very big rock, but hey, baby steps.
(roughly in order of contribution) For anything with one unfastened end of finite mass, hanging from a finite mass in a very inclined orbit,with a relatively large moon nearby, all in an elliptic/eccentric orbit around the sun, and considering Jupiter,in a gravitational field gradient,... lelz. nooooo..! Too nonlinear. Driven oscillations in the tether would eventually destroy the system. I'm unaware of any practical material and/or structure that would make it possible. You'd have to make it incredibly wide (or that infinitely rigid unobtainium) to damp out the oscillations. Even for the equatorial-geo-asteroid-space-elevator, with one end of the tether attached to the Earth, there will have to be considerable flexibility built into the system, and things will (of course) be broken into smaller pieces for assembly on/in the asteroid. Combating damping will be a monumental challenge, let's just assign it to Devac for homework.
...really, they wanted to put some cool pictures together. I for one applaud their audacity.
I'm throwing my Rotten Tomatoes (is.. is that where that comes from?).
P.S. maybe one end of the tether could generate calculated optimal destructive interference movements while the other is loaded/unloaded at a stationary point. But that would have to be a huge facility, considering scale sizes.
P.P.S. if you devised a way to steadily beef up the structure around a space elevator, outwards, gradually boosting the asteroid orbit while keeping the system's center of mass at geostationary height... hmmm...