I read the edition of the questions and they can easily be answered together. The two questions are, in a way, one.
First, the geostationary orbit is not a point where objects will be completely still from the point of view of an observer on the Earth's surface. At true, the orbit is called stationary just because the variation is just a tilt. There are always gravitational influences from all other bodies, such as the Moon, the Sun, Jupiter, Mars and even the satellites that share the orbit, in different amounts, which cause small variations that those bodies will be subject to. Even variations in the density of the Earth's interior cause minor influences, Arthur C. Clarke in The Fountains of Paradise mentions this as a justification for building the elevator of the novel in Trabopana and not in Africa or South America. A body orbiting the equator at 35.789,2307 kms from the surface at a speed of 3.074,89 m/s will describe a movement that is too short to be noticed. These variations in the Earth's gravitational field, added to the interference from other bodies, make it necessary to orbital correction from time to time. Orbit corrections in geostationary satellites need to correct their speed by ~50m/s per year.
When lowering the cable of a space elevator, the center of mass must remain at the height of the geostationary orbit to make the need for orbit corrections as little as possible. An object many times more massive than the elevator cables would guarantee greater stability. Even a small asteroid would have several times the mass of the cables to make the variations too small to be worrying. In order to know how much time it would be necessary to make corrections, it would be necessary to know how much the center of mass would shift beyond the typical variations.
When you think of a society in which you will be able to build such a structure, it is difficult to think how they could fail in basic things, which eliminates most of the major problems. The biggest problem is some unforeseen weather condition, exceptional gusts of wind, or perhaps some solar storm messing up the ionosphere. Satellite debris or meteoroid impact along the cable can also cause problems.
Another point to consider is the speed of the cable on the lowering. While 35.000 km above the surface the structure's speed will be 3.074 m/s, at the height of the atmosphere the speed is 465 m/s. Keeping a flexible cable with the tip moving inertially will cause it move to east of the target and it will need to be slowed down steadily throughout the descent, in addition to considering the conditions of the atmosphere. This will need a reasonable amount of fuel and, well, one problem that can happen there is having less than was necessary. Likewise, the tip of the outer cable will need to accelerate.