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I think the answer to your questions is differential equations. The math has been known for quite some time. Newton/Leibniz in the late 1600’s worked out the math.

You can predict where the object will be. Same way we get to the moon, or Mars, or any other celestial body.

I won’t (can’t) address relativity and interstellar distance. Length contraction bends my brain in bad ways.

Basically, lead the target. Shoot for where it will be.