let’s say you were to build a rigid ring, ten feet off the ground all the way around the planet, and kicked the supports out all at once. would the ring stay in the air? it would be pulled (almost) equaly all around the planet, and, not being able to flex, it should just stay there right?
you could compensate for varied densities in the earth by making the part on the oposite side of the earth bigger.
would it work?
Since the force on any given segment would be equal and opposite to the force exerted on a segment on the other side of the planet, the net force should amount to zero for all segments. So, yes, it should work 
Yes, it should just stay there. All the forces on the ring will cancel out. But if it happens to drift to one side, there will be no restoring force - the ring is free to drift along the plane of the ring. So you’ll need a little control mechanism if you want to keep it 10 feet above ground. Just like the Ringworld depicted in Larry Niven’s novel.
Practically, no material in the world is strong enough to make that ring. The ring is basically an 8000 mile long bridge, 5000 times longer than anything yet achieved. Also, you’d need more altitude if you don’t want the ring to hit any mountains. However, if you put it 10 miles in the air, put jet engines on it and spin it at Mach 20 or so, then the gravitational force will be cancelled by the centrifugal force. To counteract friction, the jet engines must be operated continually, fed by hypersonic tanker aircraft - we don’t have this technology quite yet, but we’re only a few decades away from it.
It’d be even easier if you put it 300 miles in the air. It’d be just like Saturn’s rings. If you make it out of sealed pressurized metal boxes, you’ve got one hell of a big space station. You can even lower a tube from there to the ground and run an elevator, as depicted in The Fountains of Paradise by Arthur C. Clarke.