You would only weigh less on the scale going down until the elevator reaches a constant velocity. When the elevator stops you will show weighing more because the relative negative acceleration of the elevator is combined with the acceleration of gravity.

Additionally, on a macro level, the Earth is hurling through space in a certain direction and if you jump on one side, it's no different than if you jump on the other because the velocity of Earth through space isn't changing.

What does this have to do with anything that I said? None of that is relative to this situation. The only forces that are significant enough for us to perceive in this scenario are the acceleration and velocity of the elevator up and down, the acceleration and velocity of the person relative to the the elevator and the Earth, and the force of Earth's gravity.