If my turn starts and my ship is on an obstacle, do I still roll a Damage die and lose my activation phase/

If your maneuver template touches some of the obstacle as you move off, then yes, you roll more damage and lose your activation phase again.

I'f you're going to hit an asteroid, be going fast enough to go THROUGH it.

If my turn starts and my ship is on an obstacle, do I still roll a Damage die and lose my activation phase/

You only get affected by it if you bases touches it when you finish your movement, or if your movement template touches it. So If you landed with the front of your base not over the asteroid, you will be fine. However, if your movement template is over the asteroid because you only just touched, it then you will hit it again and lose your action.

If you can't avoid an asteroid, fly past it.