A creature that's both a Skeleton and a Zombie will get the bonus only once.

Death Baron doesn't normally affect itself. If you manage to turn it into a Skeleton, however, then it will give itself +1/+1 and deathtouch.

Because damage remains marked on a creature until it's removed as the turn ends, nonlethal damage dealt to a Skeleton or Zombie you control may become lethal if Death Baron leaves the battlefield during that turn.