Is this "in general" thing the case with the galactic core? Or does the galactic core get special rules here, and if so, what are the nature of the rules?

Additionally, you didn't directly answer my question about the nature of force with distance: does gravity behave more like sound, where being farther makes things quieter, or like a rope, where it doesn't really matter how far you are (given a massless rope)?

From your analogy, I am no closer to understanding what you're talking about with enough specificity to be able to calculate anything.