Why is time on earth slowed down from the ship’s perspective? Shouldn’t it be faster?
According to special relativity, all non-accelerating frames of reference are equally valid, so the observations are symmetric: both Earth and the ship see the other moving away at 0.5c, so they both see the other slow down.
Now it’s true that if the ship turned around and returned to earth at 0.5c, it would be the ship’s clock that was behind earth’s, and not the other way around—but that’s because, when the ship turns around, it accelerates, and while it does so the whole non-accelerating frame of reference thing goes out the window. After it finishes turning around, the point in earth’s timeline the ship judges to be simultaneous with its own will have jumped forward in time—so that even though it observes earth-time moving slower than its own during both the outbound and return trips, the time jump as it turns around will more than compensate.
(Or equivalently, you could say that after turning around, the ship observes its own past history on the outbound trip to have been slowed down even more than earth’s.)