It would appear to me that there’s not really an “at rest” state in terms of movement in space. We’re on a planet orbiting a sun that orbits the central point of our galaxy, which in turn is also moving and perhaps orbiting something else even bigger. So, “at rest” is always subjective.

Correct, that’s classical relativity, from before the notion of spacetime even existed.

But the twin paradox says if a twin travels away at a high speed then returns, they would have aged less.

It depends on who does the acceleration. Acceleration is key for the twin experiment, and the reason it works is because they are the one accelerating and decelerating. This requires “general” relativity to make sense, it can’t be explained by merely special relativity.

Whatever is influencing us, there must be a 3d vector in space we, as a whole, are moving toward (probably constantly changing though). So, surely if you sent someone in the opposite of that vector, time for them would actually speed up rather than slow down.

I don’t get what you mean here… which 3d vector are you referring to, and why should it exist? One of the important concept is that every reference frame is equally valid.

Would they not “catch up” and be on part overall with time passing?

From one pov the earth sped up, instead of the twin’s spaceship slowing down. These are both valid interpretations. You can always catch up by doing the opposite of what you did before, but you need to look at acceleration for it to make sense. Acceleration is what turns your 4d vector so your reference frame changes, and you can sort of arbitrarily change “passing of time” if you can induce unlimited acceleration.

We wouldn’t see this skew on satellites and the like because they’re travelling with us on that vector but at a constant speed relative to us too. But sending away from us and back, we probably would.

I feel like you’re rediscovering the idea of an “aether” which was disproven by Lorentz and Einstein.

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Note that I’m not a physicist, just someone who loves staying up to date with modern physics!