Earlier quoted context omitted.
I was going to bring this concern as well. I expect that in theory we could build a space elevator but I doubt we could keep it in space. Between space junk, meteorites, and regular gear flying around like GPS nodes, items in a non-orbit (which they would be in this configuration) will be hit with more than enough energy to destroy even the strongest materials we have out there.
I would also want to know the impact of a counter weight attached to the earth and it's effect on our ability to orbit naturally while it's drag is attached to even water. Could an impact on this force it into a spin around Earth creating Centrifugal force or something not studied before?
Actually if you look at the orbital mechanics, it would offset the center of mass for the earth in relative proportion to its mass. That said, the Earth has a lot of mass and so that effect would certainly be imperceptible and probably unmeasurable (although things I used to think we'd never measure directly we seem to do so now). It would do any "damage" in the sense you are using the concept.