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According to Hubble more distant galaxies are moving away faster but since the light took billions of years to reach us we should say they WERE moving away faster billions of years ago. How do we know that those galaxies where light is just reaching us know are STILL moving away at all.... let alone faster

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We don't know. This is a depressing thing that makes me, at least, feel so small. The only thing we might assume, that if they once were accelerating that fast, they probably will do it still (even faster), but of course we can't be sure if anything have happened.

It's like this with all the long distances in the universe. How do we know if Proxima Centauri (4.2 light years away) have died/exploded? We wait, 4.2 years.

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I think you raise an interesting point that is usually glossed over, and it's really a cosmology question (since it concerns the large-scale doings of the universe). And I think the best answer is that we assume the universe to be 1) isotropic and 2) homogeneous, meaning that if the universe itself has some property somewhere, then it has it essentially everywhere. These cosmological assumptions, known as the Cosmological Principle, are very well motivated: see for instance here, which explains that so far large scale experiments justify the assumptions.

So in truth, we don't know in the sense that I know what time it is on my wrist watch; rather, we know it in the sense that I know the sun will rise again tomorrow - it is a well verified assumption about nature's doings.

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