Can I legally kill such protected wildlife as woodpeckers, swallows, owls, or robins if they are causing damage to my house?

The federal Migratory Bird Treaty Act protects woodpeckers, swallows, owls, and robins. Any permit to lethally control these species would be issued by the U.S. Fish and Wildlife Service and only in extreme situations. See WDFW's "Living with Wildlife" fact sheets for more information about these species.
A permit is not required to remove woodpecker, swallow, owl, and robin nests abandoned after breeding season, or nests under construction that do not contain an adult, eggs, or young.