If my movement would cause me to overlap an enemy ship that is also on an obstacle like an asteroid, but after I reduce my movement I am no longer on it, do I take damage from the obstacle because I would have overlapped the asteroids in my original move, or do I just take the collision damage for overlapping the other ship, and not the asteroids?

Conversely, if there's an asteroid between me and another ship, and my speed 2 maneuver overlaps the enemy ship but not the asteroid, but slowing down to speed 1 DOES put me on the asteroid, do I take a face down for the collision AND a face up because I'm now overlapping an asteroid, or do I only take the face down because my original course didn't overlap an asteroid?

You look at the real end position of the ship.

So if reducing puts you on an asteroid...slam in da face:)

Vitalis is right. The rules are as follows:

You check for overlapping ships and resolve the appropriate effects (reducing speed and taking a facedown damage card) while executing a maneuver.

You check for overlapping obstacles and resolve the appropriate effects (faceup card from asteroids, 2 damage from debris, or 1 repair from the station) after executing a maneuver.

Correct DiabloAzul.

Final position matters for this.