I envision a conductive wire that is extruded out of a satellite as it completes it orbit, and then goes past it's start point a bit before stopping the extrusion. Then the satellite is sent to grab both ends and connect them into a complete wire ring around the earth, let's say in the altitude range of geostationary satellites. Could do this many times and create a cable. This ring is moving faster than the earth rotates, maybe several times faster. If you then put a satellite in geostationary orbit that can put a wire loop around this earth-ring, couldn't this generate electricity to be sent to earth?
I'm not quite sure what you are proposing. A wire ring around the Earth (assuming you could get the necessary quarter million or so kilometers of wire up into orbit) is fine. If it's very strong wire it could be rotating a bit faster than orbital velocity, held down by the tension in the wire, but (a) no real material would let you go much faster and (b) it becomes orbitally unstable if you go too fast. However this doesn't do anything about generating electricity. If you run a current through the wire (which we may as well assume is superconducting) it will act as an electromagnet, adding to (or counteracting) Earth's magnetic field.
Now, when you have a wire moving across a magnetic field an electric potential will be created, and, if the wire is part of a circuit, current may flow in it. If current does flow, then the wire will experience a force, tending to slow its motion relative to the magnetic field. So, you can do this just using Earth's natural field (look up Electrodynamic tethers), although not around geostationary orbit, because a satellite there is not moving relative to Earth's magnetic field. However this doesn't make energy from nowhere. You are converting kinetic energy of the satellite into electrical energy (or, you can do the opposite, which is often more useful). Since that kinetic energy came from the rocket fuel used to launch the satellite, which took lots of energy on Earth to make, there is no net gain.
It wouldn't work in equatorial orbit - because at that angle to the magnetic field no power would be created. It work in polar orbit - for a very short time.
Making the ring turn around it axis would make each bit of it behave like a satellite, preventing it from falling to Earth - while its own rotary motion would not cause any electric current and exceeding the orbital speed a little bit would keep the ring taut, plus its rotary momentum would make it maintain the same orbital plane.
Meanwhile, Earth, turning under it, and its own magnetic field turning with it, would generate electricity in the wire. Too bad, along with electricity it would create electrodynamic force which would act upon the wire pulling it sideways - and the forces involved would be sufficient to rip it to shreds within moments of closing the loop.