Yes - you can walk your dog in the area surrounding the national cathedral (I used to live in that area). However I do not think you can bring your dog inside the cathedral unless it is in a bag or small crate. Mainly small dogs are allowed in places as long as it is not a business that sells food. If you bring your dog inside the cathedral - you may want to do it while there is no service or event being held. It could be embarassing if your doggie decides to bark at the wrong time.