Misconceptions in Relativity

According to General Relativity (and most related gravitational theories) gravitational effects propegate at the speed of light. This is consistent with the limited number of experimental/observational tests carried out.

I'm fond of pointing out that if you invoke magic to set up your experiment, you cannot expect meaningful results! You cannot simply vanish away the mass of the Sun using any physics of which we are aware.

However, answering in the spirit of your question, GR is a *local theory*. Space-time geometry at a point depends on the mass/energy distribution at that point. There is no action at a distance. The space-time curvature 7 light minutes out from the Sun is not a direct result of the Sun's mass, but of the space-time curvature slightly less than 7 light minutes out. Think of the old rubber sheet analogy. So the space-time will relax outwards from the centre, and will take eight minutes to affect the space-time around us.

We use geometry - topology refers to the connectedness of a space, and is a whole new subject within General relativity - but yes, that is the idea. However, it is very difficult to test directly.

It can't

Because you can't magic mass/energy into and out of existence. All you can do is shift it around. And I don't need to tell you that there is a limit to how fast you shift mass/energy around. Hence the problem...

All you can do is look (indirectly) at gravitational waves. These do definitely appear to propegate at the speed of light.

Not really. If the Sun collapsed to a black hole today, what gravitational effect would this have on us?

No. The gravitational field here would be just the same as it was. The event horizon of that black hole would be (pure guess here) a few kms only. Outside that we'd just keep orbiting like we do now.

*We* haven't - we are a long way from the Sun. All we care about is how much mass/energy is contained within our orbital radius. We don't give a damn how dense it is, just so long as it is a spherically symmetric distribution. Blow the Sun up, and as long as the entire debris cloud remains within our radius, and remains spherically symmetric, we'll keep orbiting. Likewise, should the Sun collapse. The only difference the collapse will make is that we can know get much closer to the interesting parts of the Solar space-time: r=2M (the event horizon, about 3km for the Sun), and r=0 (the singularity). Normally, we cannot encounter these regions of space-time as they don't exist, as the Sun occupies that space.