So you make your maneuver and end up on top of an asteroid. You skip your action and your attack and you roll a dice to see if you take a point of damage.

The next round, when you move to leave the asteroid, your maneuver template overlaps the asteroid still. Do you have to roll another damage dice and skip your action again?

Yes.

If the manoeuvre template overlaps it, or you bump someone and land on it again, you take its effects again.

As mentioned, if you hit it again you suffer its effects again.

While probably not 100% accurate you can think of the area underneath your ship as being completely clear at the start of the activation phase but after the ship activates it is then whatever it is supposed to be. If you can move off the rock without the template touching it then it's like you were never on it; clip it with the template and it's just like you ran over a different rock.

Cool. Thanks!

So you make your maneuver and end up on top of an asteroid. You skip your action and you attack and you roll a dice to see if you take a point of damage.

Quick clarification:

Above it says, "and you attack" does this mean the ship that landed on the asteroid attacked an enemy ship in this game?

If so, that should not have happened. Landing on an asteroid takes BOTH the ship's action and attack away for the turn.

See page 13 of the rulebook.

Quick clarification:

Above it says, "and you attack" does this mean the ship that landed on the asteroid attacked an enemy ship in this game?

That's a typo. It was supposed to say "and your attack".