Q: "Just how close can a Warship get to an airless world?"
A: A few tens of meters below the surface, depending on speed at impact.
In theory, one could come close enough to an airless world to scratch the paint, but the amount of structural stress that the maneuver puts on the ship may create a practical limit, depending on the strength of the gravitational field. For a small moonlet, not a problem, and you can probably come close enough to reach out and touch the surface (if the pilot is REALLY good and you don't mind having your hand sheared off), but a higher gravity planet might break your ship apart, between the gravitational pull and inertia.
Then again, to pull the kinds of maneuvers that a warship does, the structural integrity has to be absurd, with anyone near the ends of the ship being crushed by the centrifugal forces as it spins in place. The ship might survive the maneuver, even if the crew doesn't.