I Love these Ideas.....
Little hassle with rotational and gravitational physics....
For all of these components to work in space, they all need to be located in the same geostationary location.....
As the tethers between all of the components will be "infinitely" long, they will have infinitely low stiffness (relative to their length.
Lower orbits have a higher velocity needed to maintain the orbit... Higher orbits (further from the earth) have lower velocity needs to maintain in orbit....
Ok putting the main station in Geo-stat orbit is cool... Then putting the counterweight a million more miles out into space (hyperbole) and keeping it rotating at the same surface speed as the earth.... it is doable, and produces the tension needed to keep the tether "tight" as the mass will be trying to fly off into space.....
The outer counter-mass will be travelling orders of magnitude higher velocity than the necessary orbital speed (speed through space) as it must "maintain pace" with the base location on earth..
The whole thing will need rocket boosters (lots of) to keep the mass positioned properly.. (stopping it from slowing down and dragging the whole thing around the earth like a huge piece of wrapping string.) and if you were wanting to sweep all of the space junk out of the orbits between the station and mass, it will do that too (sure not many applications put satellites further out than Geostationary, but there may be some).... as well as potentially knocking out any satellites which some idiot put in one of the in between orbits.....
As I said, Love the idea, but in all probability rockets will be cheaper for some time to come. Lets focus on cheaper and more environmentally friendly rockets, ion motors, linear accelerators (for micro satellites) etc.

I am not saying the people behind this are crooks, but they don't know what they are doing.
They seem to have failed to notice that the moon is tidally locked with earth so the moon spins on its axis once every 27.3 days or so. This does not make constructing a tether impossible though. If you attach the tether to the point on the moon nearest the earth and run it through the Earth-Moon Lagrangian Point 1* to far enough into the earths pull so the counterweight pulls the tether tight lets call it a minimum tether length of 60,000km. To have a low energy docking with the tether high point station it will have to happen at the L1 and the tether docking station will have to constantly clime up and down the tether or be readily accessible only once or twice every 27.3 days or so. (See below)
Most importantly building the tether is a waste of time, energy, material, and effort. Just barely pass L1 and and you will fall the rest of the way to your destination weather it's the earth or the moon. The moon effectively doesn't have an atmosphere so there is no reason not to build a linear accelerator on the moons surface that if engineered correctly will also able to catch incoming vehicles as well with most of the installation below ground where it is safe from most of the hazards of space. The accelerator will also be able to throw stuff to deep space by just giving it more velocity and firing at the right time in the moon's orbit.
*Earth-Moon Lagrangian Point 1 the point of gravitational equilibrium between the earth and the moon. The moon being in an elliptical orbit makes this point of gravitational equilibrium very in distance from the earth and the the moon even though it remains at the same percentage of the distance from the center of the moon and the point where the moon and earth orbit each other. This point is about 17,000km below Earth's surface but this is not at the center of the earth.

re; MD
Actually the higher the orbit the faster the object has to be traveling but the distance it has to travel goes up faster than the velocity needed for the given orbital altitude, so that it takes longer for the higher orbital satellites to cross the sky above a ground observer.
The Freefall Station (At the hight of geostationary orbit) will consist almost entirely of freefall labs and factories, and docking for returning spaceships such as asteroid miners looking for a market and dive to let of steam. It may prove quite profitable to move ships from Freefall station to highpoint station charging less than the reaction mass would cost to give the same velocity change.
If you take a string and tie a weight to one end and and a loop around the other and placing the loop loosely around your finger and then swing the weighted string around your head you do not need to put rockets or any other engines on the weight or string to keep the string from wrapping around your finger. The same principal applies to the beyond geostationary counterweight on an orbital tower.
The orbital space particularly below geostationary orbit will have to be cleaned up as part of the construction process and then it will be up to the owners of new LEO satellites to avoid the tower.

Pikeman, Kepplers laws might be worth a read - the moon orbits the earth at an altitude of 210,000 miles, but has an orbital velocity of only 2,412mph, meanwhile, if a satellite in low earth orbits at only 500 miles it has an orbital velocity of 16,920mph. http://armycommunications.tpub.com/SS0031/SS00310027.htm
Also, in an elliptical orbit, the perigee (lowest altitude) is the point of highest velocity, and the apogee (highest altitude) is the point of lowest velocity.
Kepplers laws are the constant bain of those attempting to design high-resolution spy satellite networks, because it's impossible to achieve a geostationary low earth orbit!
A space elevator for the moon is much more feasible than on the Earth - much shorter cable or ribbon required, and no atmospheric drag. A space elevator for use on Earth? - Good luck.

One excellent reason to build a space elevator on the moon is to reality-test the concept. We can only guess at how a space elevator will behave, there has never been anything like it built.
That said, it makes sense to test it on the moon, where the concept can be scaled down to fit lunar gravity, where it won't interfere with terrestrial satellites, and where there is no biosphere to screw up if something goes very wrong.

1st....why? Moon mining? Will the next thing for the sheeple to cry about is "keeping Luna grey!"?
2nd...by 2020? Really? LOL...umm, no. Just, no. It's less than 8 years 'til 2020 rolls along, theres absolutely no way this happens in that time span, none at all.
3rd...anyone who can afford to Kickstart to such a frivolous endevour CLEARLY isnt being taxed enough & needs to have all thier wealth redistributed to people...er....who arent as "fortunate". There's ghetto kids who can barely keep thier smartphones paid for, to think money would be spent on such a project should send your college professor into convulsions.

The point is to get us off the Earth cheaply. Without that, doing anything on the moon will be prohibitively expensive. We need to dump all these pie in the sky ideas and focus on technology we have that can be enlarged to get us off the planet cheaply. One Large linear accelerator up the side of a mountain to launch vehicles and payloads which utilize small reusable rockets would make manned deep exploration a reality. The technology will soon be used to launch aircraft from carriers. Granted, one for space would be a thousand times longer. And use a Maglev sled for the vehicle/ But we would just need one.....

The elevator I'd like to see is a bit different. It'd be much shorter, about 1/5 as long as a regular space elevator, and it would 'walk' or 'roll' around the earth, stopping at various 'ports' along the way. Of course, it would require a means of quickly loading/unloading the 'car', but that should just be a matter of technology.

It is entirely possible that our financial problems could be solved by mining and manufacturing in outer space. While the dream ideas here are nice to discuss, we need to look at more realistic solutions that are available now before we drive ourselves into a financial pit. Our economy is in grave danger, but not beyond rescue if we think outside the box. The mineral wealth of the Moon, Mars and the Asteroid Belt could bring us all new and lasting standards of living better than ever before. Doomsayers have always been around, but so have opportunities like the mineral wealth off planet Earth.

The article author needs to look up the difference between geosynchronous and geostationary.
There are many possible geosynchronous orbits. They don't even have to be perfectly circular. They're called synchronous because the orbital period is the same as the rotation of Earth or averages out to the same. A satellite in such an orbit may drift up and down or back and forth a bit, tracing out a horizontal figure 8 pattern.
There is only one geostationary orbit and it's right over the equator.
A space elevator could have its top in a circular geosynchronous orbit but it would add to the complexity to accommodate the movement.
As for the counterweight it might be possible to use electric coils to thrust with/against Earth's magnetic field to keep it in place directly above the top of the elevator.
For a moon to L1 elevator it wouldn't be scaled down it'd have to scale up, at least in height since L1 is nearly 2x the altitude above the Lunar surface than geostationary orbit above Earth's equator.
Double the material but 1/6th the gravity.
Theoretically a moon elevator could be run out to L2, L4 or L5 but they're a heck of a lot farther away. Probably be "easier" to spin up the moon so the elevators could be no longer than an Earth one. ;-)
May as well build elevators on Venus to draw off 70~90% of its atmosphere and spin up its rotation to 24 hours. Without the technology to do that, forget ever terraforming Venus.