Yes, dark matter does influence normal matter–gravitiationally. It is extremely important for how galaxies or systems of galaxies form and interact, because as far as we can tell galaxies are much more massive than we would expect based on the stars and dust in those galaxies.
That’s because, as I said, dark matter appears to interact gravitationally with normal matter, but not in any other way. Dark matter is concentrated in galaxies, but it isn’t particularly concentrated in solar systems. So stars and black holes and dust clouds and dark matter orbit around the center of the galaxy. The dark matter passes through our solar system without interacting with it.
So remove the galactic dark matter and what happens? Exactly the same thing that happens when we remove the normal matter from the galaxy. We just lose the gravitational effect. And as was shown above, that gravitational effect is very small. Sure, the galaxy (include all the dark matter) is very large and very massive. It’s also very far away and very diffuse.
So yeah, dark matter affects us via gravity, but in no other way. The reason we know that is because if dark matter interacted in other ways, we could detect those ways. Since we can’t, we know it doesn’t. Or rather, if it does, it interacts so very weakly and rarely that we haven’t managed to detect any of those interactions yet, which puts an upper bound on how strong those interactions could be. We know they can’t be strong or common, because if they were we’d see them. So the interactions have to happen so rarely and/or so weakly that we haven’t managed to detect them over the last 200 years of experimental physics.
Again, we’ve detected gravitational interaction, but nothing else. And we know our solar system isn’t particularly concentrated in dark matter compared to interstellar space because if it were we’d find that every object in orbit around the sun would act differently. Since dark matter doesn’t seem to interact with normal matter except via gravity, it doesn’t clump into planets or stars. A normal matter particle falling into a planet gets stopped by the electromagnetic forces–it hits the ground and is stuck on the planet. A dark matter particle just falls through the planet and swings on out the other side in a hyperbolic orbit, just as if the planet’s surface doesn’t exist.
Obviously, if we have alien space bats that can ping the entire rest of the universe out of existence, we have no way of saying what the laws of physics in the remaining universe that consists only of our solar system will be, they will be whatever the alien space bats say. But as far as we can tell, what we call dark matter is only an important influence on galactic scales, and what we call dark energy is only important on intergalactic scales. Since those won’t exist anymore after the universe goes “ping”, they won’t matter. Unless the alien space bats say they do.