Rocket acceleration is less than 3g at all times.
Breaking under its own weight is a different story entirely. However, if you use nylon, this problem gets a lot simpler.
Granted, Col Armstrong is referring to different situation than yours, but I just thought that it was funny...
Capturing an asteroid is difficult and nearly impossible. Let's face it, if a huge asteroid comes that close to Earth that we can capture it, we are not going to be like "oh it's a friendly asteroid that we can use to build a space elevator". It's going to be like "My fellow Americans, we are...
It's obvious that this theory has some flaw in it. All the "space elevator" concepts out there depend on one thing, a rigid wire which is held in orbit by a large mass. This large mass is critical because it holds the string taught.
Let's assume for a second that your idea works and you get...