does a ship with the stunned pilot critical card have to roll for damage if the ship through either another ship or asteriod where the base does not land on the other ship base or asteriod?

First of all, note that with the Stunned Pilot card you do not "roll for damage" - you suffer 1 damage regardless. This may remove a shield token or result in a face-down damage card if there are no shields.

Second, "overlap" only occurs when the ship's base in its final position would overlap either another ship or an obstacle token. In the case of ship overlap, your ship is moved backwards, but "overlap" has occurred and Stunned Pilot comes into effect. For obstacles (asteroids), the ship remains on top, overlapping the obstacle. If only the template overlaps, but the ship base does not, then I do not believe Stunned Pilot would be in effect. Would anyone argue, in the case of obstacles, that template-only overlap would trigger Stunned Pilot?

Note also, in the case of obstacle overlap, that you still have to roll for damage as usual, and the Stunned Pilot damage is in addition to any damage sustained by rolling.

Given that the damage roll for an obstacle triggers if only the template overlaps it, I would say that Stunned Pilot would indeed also trigger, if for no other reason that to keep the rules as simple as possible. The rulebook is kinda vague on what exactly "overlapping" means, but the intent seems to be that overlapping ships only the bases matter, but overlapping asteroids the base and the template matter.

From the FAQ, Page 6:

If a ship with Stunned Pilot assigned to it executes a maneuver in which the maneuver template overlaps a ship or obstacle token, but the ship’s final position does not overlap a ship or obstacle token, the effect of Stunned Pilot does not resolve.

Thank you all for your help.

Well alright.

As usual, Buhallin can quote Chapter and Verse...