Not only is there no universal up and down in the universe, there is no forward or backward, and no left or right. There is also no universal reference to measure your speed.
All of these measurements are relative. They have no meaning until you designate a reference from which you are measuring. Once you declare that “that way” is up, then the other way becomes down. If you declare that the sun is stationary, then the speed of everything is determined by measuring how fast things are moving towards/away from the sun. But if you changed the reference to be the center of the milky way galaxy, then the speed of everything would suddenly be different.
Even in your question, we customarily think of the north pole as “up” and the south pole as “down”, mostly because we’re used to seeing maps and globes depicted that way. But there’s no objective up and down in the universe.
Regarding your specific question about our solar system, all of the planets formed in the same general plane, so if you were on the north or south pole and you looked radially outward from that point, you would not see any planets. All of the planets are basically lined up and moving in the same plane, because that’s how solar systems generally form.
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