Could large craters on the moon be used as reflective lenses for radio signals?
Acting like a large radio telescope reflecting radio waves to a satellite positioned over the crater.
Interesting idea. I think the answer is both yes and no -- yes with a manufactured dish but no in the crater's raw state.
The Arecibo telescope sits in a natural crater, but adds a dish which has a couple of important things required by a radio dish:
Edit: Added #4
You'd have to line the surface with something reflective to microwaves, like a metallic mesh, or similar materials.
Secondly, the shape of the crater is probably not quite ideal, so it would have to be adjusted a little, carved up a bit in various places. But it's a good start, and definitely better than starting with a flat ground.
There is also the question of stability - you need to make sure that whatever changes you make (carving a different shape, lining it with mesh) do not affect the stability of the crater, or else various parts may slide or collapse. This is an engineering problem.
Not possible unless the crater is exactly on the equator, and even then it would be tricky.
But a crater like the one in your picture is so strongly curved, the focal length is about the same as the diameter. In other words, if the diameter of the hole is X, the altitude of the receiver is pretty close to X - give or take something like 50% or so, depending on the exact curvature. It might be easier to just build a giant arch over the crater. Again, this is a matter of engineering.
I agree that the crater needs to be lined, but you also have the problem of maintaining a satellite in stationary orbit above the crater. Nearly impossible unless the crater is on the equatorial plane. Also, a stationary satellite around the moon would be influenced by the Earth, so you would need to burn fuel to keep the satellite in position.