Will the station at the end of the tether still need a rocket to deal with the additional mass? Is the fact that this rocket only has to go up once the core advantage?
Wouldn't the load going up the elevator pull the tether to one side?
There are more reasons to build a space elevator, but that's a big one. It really would be far more energetically efficient.
To get near 100% efficiency, it follows from the kinetic energy formula and conservation of momentum the body you're pushing against must have a much larger mass than you[1].
So discounted exotic phenomena, the only way to travel efficiently through space is to push off a very large mass, departing at full velocity, or in the case of near-Earth travel, simply push Earth.
The space elevator is essentially an elaborate staircase. It is in a stable equilibrium, and climbing it doesn't steal energy from the counterweight (which in fact doesn't move in fact because it is in constant tension); you're just pushing Earth away.
In the grand scheme of things the very low efficiency of maybe 10% (in my guesstimation) to LEO isn't so bad (not even the maybe ~5% interplanetary efficiency); especially considering the costs of space systems in general in comparisson to fuel cost. This small fuel cost makes reusable rockets quite an attractive option near term (as noted by SpaceX).
But long term, that's quite a steep inefficiency. If we were to endeavor large scale colonization or exploration of exoplanetary resources it seems to me either a kinetic launch system or at least a space elevator variant would be a necessity.
[1] Derivation: M1 v1=M2 v2 => v1^2 = (M2/M1)^2 v2^2; M1 v1^2 + M2 v2^2 = E => M2^2/M1 v2^2 + M2 v2^2 = E; M2 v2^2 (1+M2/M1) = E => K2 = E/(1+M2/M1).
As M1->infinity, all the kinetic energy goes to K2 and none to K1 (which is why you don't give Earth any meaningful energy by walking).
The tyranny of the rocket equation! The energy use of a rocket isn't just "how much energy is needed to have that kinetic energy and that potential energy", but also includes the energy needed to lift and accelerate the fuel used to provide that energy, as well as the fuel needed to provide that energy, as well as the ... and so on.
The rocket equation is about reaction mass that is carried by the thing expelling it. It does not apply to climbing a rope, nor does it apply to the flight of a helicopter.
Regarding pulling the satellite down,
Pulling down on the weight at the end is something we have to do anyway to keep it from flying out away from earth.
Think of it like holding a rope tight between your arms and putting a robot that moves on it but ends up not exerting enough force to cause it to slack. Your arms (earth/space teather and the in space counterweight) and the object being moved would keep the same overall motion between all 3 objects (your body would just turn).
The space elevator is similar except earth is way more massive than the object being moved so nothing really changes. Also, due to gravity, you have to expend extra energy to move from the bigger object (earth) to a smaller object (space elevator counterweight), aka, the minimum energy cost of moving things out of gravity wells.
Maybe the required energy would be sucked out of the earth's rotation, but that seams insane.
As you climb the cable, the force of gravity pulling you back to earth decreases, and the centrifugal force pulling you away from earth increases. The difference between these two is the force you need to provide to climb the cable.
I believe you are correct for tethers much shorter than geosynchronous orbit. Below geosynchronous orbit, the force of gravity is higher than the centrifugal force. Therefore, an object climbing a space elevator will have to provide energy equal to the integral of the difference between the centrifugal force and the gravitational force across the distance traveled. The remaining energy (the remaining gravitational potential and the kinetic energy of the orbit) will be leeched from the orbiting counterweight (requiring the counterweight to have a rocket to maintain orbit, as you suggested)
For tethers that extend beyond geosynchronous orbit, it is possible to for no energy to be removed from the counterweight (instead, all the non-climbing energy will be taken from the rotation of the earth). Imagine that we place a counterweight on a tether beyond geosynchronous orbit. This counterweight and the earth it form an orbiting two body system. The tether will be under tension (the force necessary to keep the counterweight in synchronous orbit) -- let's call that force T. A climber that scales the tether will exert some force T_1 on the counterweight, pulling it towards the earth. However, as long as T_1 is less than T, the counterweight will remain where it is. The force of the table on the earth will become T_2 = T - T_1. In other words, a portion of the force necessary to keep the counterweight in orbit will now be applied by the climber instead of by earth. The energy that the climber must apply is the same as before, but the counterweight is not affected. The remaining energy, by process of elimination, must come from the rotation of the earth.
Geosynchronous orbit is 42 km from the center of the earth while the ISS orbits 7k km from the center of the earth. I expect the experiments are being done at the ISS for convenience rather than from a plan to build a space elevator to the ISS. The article also cites speed and distance numbers that imply reaching a geosynchronous orbit.
To answer your questions more directly: 1) below geosynchronous orbit, yes. 2) No, the ability to extract energy from the earth's rotation is the main advantage. 3) Yes, but for a counterweight beyond geosynchronous orbit, the tension on the tether will pull it vertical.
s/center/surface/g