You're asking about an idea that's been around for a long time: the Space Elevator or the Sky Hook. Konstantin Tsiolkovsky wrote about a similar concept in 1895, though his concept was for a standing building, a compression structure supported by a foundation on Earth.
You can't just lower a cable from a geo bird. If you have a cable hanging from a geo station with nothing above it (ignoring the dynamics of getting that cable there), the weight of the cable, not fully offset by centrifugal force at the lower altitudes, would pull it down. You have to send a cable outward too, such that the mass and increased centrifugal acceleration of the outward segment balances the mass and decreased centrifugal acceleration of the inward segment. Once the inward segment reaches Earth (Sri Lanka was A.C. Clarke's favorite) it can be anchored and a counterweight attached to the outward end to keep the whole thing in tension, even when a payload is being hoisted up the cable.
That's the general attractiveness of the concept: if you could build such a thing, you could use it to "elevator" payloads up to GEO using only electric power. You could also return things from GEO to Earth for such things as repair or servicing.
Until fairly recently there were no known materials that could handle the tensile stress involved. But it appears that maybe carbon or boron nitride nanotubes, or diamond nanofibers, might be strong enough. But the static tensile stress is only part of the problem.
One problem is the "bullwhip" problem. A payload being sent upward places a sideward force on the cable because its tangential velocity is being increased, ultimately to GEO orbital velocity. This launches a displacement wave traveling both upward and downward in the cable. The wave traveling in the direction that the cable thickens isn't too much of a problem, but the wave traveling in the direction the cable thins grows in amplitude as it travels. This is the same phenomenon that allows someone to crack a bullwhip: the wave launched at the handle grows in amplitude as it approaches the skinny end, and that end can reach supersonic speeds. As the payload nears the GEO terminus, the wave launched from that thick part of the cable travels toward thinner sections, and the large displacements from equilibrium can result in large, sudden velocity changes that greatly increase the stress on the cable.
Another obvious problem is orbital debris and meteorites. Cables under extreme tension don't ike being partially severed. If a crack or gouge is big enough that it becomes a Griffith crack the whole cable severs, and that's a really bad day.
To answer the newly-edited and emphasised part of the question: no, it would not be subject to re-entry speeds and heating in the usual sense. Construction of the cable is a carefully monitored and controlled balance of sending new cable both upward and downward, to keep the net motion of the "station" at GEO constant, at GEO velocity. Even with new high-tensile-strength materials such as carbon or boron nitride nanotubes, at the GEO point, where stress is the maximum, the cable diameter required is really large, too large to "spool". It might be built up from spools of smaller-diameter cables, or even chemically fabricated in a continuous process. But either way, the net descent rate will be low, limited by the maximum fabrication rate that process allows.
And because the upward and downward segments cancel net tangential forces as well as radial forces, this prevents the usual acceleration to high tangential speeds that you usually get when an orbiting object descends. That acceleration occurs when the orbiting object is in freefall (i.e., subject only to gravitational forces from the primary, the large body the object is orbiting), but the end of the cable isn't in freefall: it has large non-gravitational forces on it from the rest of the cable.
The net result: you don't have to worry about re-entry heating.
EDIT 2018 July 15
I wrote software to calculate the required cable diameter as a function of altitude, only for a static cable in position, so no installation dynamics, no propagating waves, and no design margin! so it gives the minimum diameter the cable could be under perfect conditions, at each point along its length. Step-by-step upward from the anchor point (and the specified bias tension there) it calculates the net acceleration due to gravity and centrifugal acceleration, calculates the differential tensile force with altitude (integrating upward from the anchor point as it goes), and then at each step uses the material's tensile strength to calculate the cable diameter. Putting in different material properties yields interesting results.
Using Kevlar, with a tensile strength of 3.62 GigaNewtons (GN) per square meter and mass density of 1,440 kg per cubic meter, and a bias tension of 1 MegaNewton (MN), the cable diameter at the anchor is ~1.87 cm, but a whopping 285 m at the GEO point. The cable mass from GEO down is ~1.2E12 metric tons!
But using carbon nanotubes, assuming you can get large-scale performance as good as the small-scale performance demonstrated in the laboratory (63 GN per square meter and mass density of 1,400 kg per cubic meter), the cable diameter at the anchor is only 0.45 cm and only 0.77 cm at GEO, with a down-segment mass of only ~2,000 tons. But this is an extremely optimistic estimate. It assumes that any individual carbon nanotube that starts out at the anchor continues unbroken all the way to GEO, and that any nanotubes added as you go upward also extend all the way to GEO, and bind so strongly to their neighbors that slippage doesn't occur.
Even if the large-scale performance is only 1/10 of the lab performance, that still reduces the cable diameter at GEO (compared to Kevlar) to ~3.1 m and total down-segment mass of ~1.7E8 tons.
Again, this is under ideal conditions. When you start adding the strength to handle orbital debris and meteorite damage, dynamics such as propagating waves, systems to control the dynamics, and the ever-present design margin (if you've ever driven across a bridge, you owe your life to that!), cable sizes get significantly larger, along with the masses.