If you were to use Unkar’s ability from a ship that has only one hull remaining (or has a nasty crit already and suffering one damage could trigger a self-destructive chain) - do you still get to do the action before being removed from the game?

Say you have a spacetug on one hull equipped with Unkar and Deadman’s Switch that bumps. Can you tractor the bumped ship before blowing up and triggering DS?

Outside of the engagement phase as soon as a ship is destroyed it is removed from the game. So no.

"DESTROYING SHIPS

A ship is destroyed after it has a number of damage cards that equals or exceeds its hull value. A destroyed ship is placed on its ship card.• After a ship is destroyed in a phase other than the Engagement Phase, it is removed from the game." -page 8 of the rules reference