Try to imagine the ring not as a stiff ring, but a series of small closely spaced asteroids loosely joined together by pieces of weak elastic.
This is because no material bond can really withstand too much tension over such an enormous distance.
So this means:
- The ring must not require any force on it to stay in place. Essentially it must be in a stable orbit.
- Being in a stable orbit just means it should obey simple orbital mechanics. In other words, yes it does spin around the Earth, and being in a stable orbit its speed of spin will be determined by the radius of its orbit.
- Would you feel weightless? Objects in orbit are in 'free-fall'. They are at rest in a gravitational field, not accelerating, in other words, you would not experience any/much gravity when in orbit. If you were on your stationary ring, you would float, because you would also be in orbit, at the same speed around the Earth and the same distance above the Earth, so you would not experience any acceleration.
Can it be a ring for a space hook? Yes, but any force on the ring must still be counterbalanced, or the ring would loose orbital speed and come apart. Therefore to maintain a stable orbit, either the ring must increase in rotation when bringing an object up, or decrease down, and conversely decrease orbital radius if bringing an object up, or increase radius if putting one down.