At that altitude and speed, the object in orbit will remain over a single point on Earth forever.
But, the author wants to move the orbiting object out to 62,000 miles. To remain stationary over a single point on the Earth, the speed of the object would have to be increased to around 17,000 mph. At 62,000 miles, this speed is triple the speed necessary to keep it in orbit. As a result, the object wants to pull away from the Earth.
This would create a tremendous force on the "overcooked pasta", depending on the mass at the end. Certainly, when that 13 ton elevator arrives at the "top", its mass is now adding to the stress on the pasta.
The part of the elevator out beyond geostationary orbit acts as a counterweight. (62,000 miles sounds too far, however.) The pull that it generates is needed to counterbalance the tendency of the part of the elevator below geostationary orbit to fall back upon the Earth under its own weight. The key fact of the elevator is that its center of mass is at the geostationary point.
Start with a small object in free-fall in a geostationary orbit. Now let it grow a bit longer. Before it gets very long, the tidal force grabs it and it points towards the Earth. It's still in free-fall, but it's under tension from the tidal force, and its rotation has become phase-locked with its orbital period. The object as a whole is still in free-fall, though.
Now let it continue to grow until it one end of it reaches almost to the ground. It's under tension, but it doesn't fall to the ground and it doesn't fly off into space, because it's still in free-fall, as a whole.
This would create a tremendous force on the "overcooked pasta", depending on the mass at the end.
But that's mostly a tidal force. There is some additional tension caused by the rotational angular momentum of the object, which also has a period of one cycle per sidereal day. That's probably enough to forget pasta as a building material, but all the rest of the angular momentum has dropped out of the equation: we've already accounted for the orbital angular momentum when we stipulated that the object as a whole is in freefall.
Certainly, when that 13 ton elevator arrives at the "top", its mass is now adding to the stress on the pasta.
Yes, the stresses change in response to the position of the carriage. I imagine, though, that the mass of the "stalk" is huge compared to the mass of the carriage plus payload, so the position of the center of mass doesn't change very much. Moreover, you could operate two carriages in opposite directions to preserve the balance.
Not true. See my post #247