I second the recommendation above. You shouldn't be paying UI or workers comp AFAIK, but yes to B&O tax and probably licensing fees both to Washington and Seattle. I'd recommend having a good contract in place.
When I was getting started I think I adapted one from Nolo, and that was fine for my needs. You could ask them (the client) to supply one and adjust it as you like.
I'm certain that both sides will want a clear contract in place before we get going. Any recommendations for boilerplate I should consult?