If a ship starts on an asteroid, does it 'move through' the asteroid when it moves out of it?

Only if the template overlaps the asteroid token. If the template does not, then you act as normal barring other outside influences.

Obviously it does, since it starts on an asteroid.

So that means you roll for damage when you land on it, lose your action, and then roll for damage again when you move in the next turn? Ouch.

No, the base and template are two different things. If your front guides are hanging off the asteroid and you place the template on the table and the template is not covering or overlapping the asteroid at all, you DO NOT roll for damage and you DO NOT lose your action.

Your base and its starting position mean diddly squat when moving off an asteroid. The only things that matter are the bases ending position and the template. The base or template needs to overlap an obstacle for it to lose actions/attacks/etc.