As long as the vet is knowledgeable on current anesthesia protocols, and takes the precautions of blood work first, before surgery, keeps her on fluids before, during and immediately after surgery, monitors her temperature, she should be fine. My vet also keeps an EKG before, during and after. Those are questions I would ask, "what is your standard protocol for greyhounds in surgery?" This surgery is external, it is not going into the abdomen, or any organ, or any bone.
I can't answer the question on extra cost. Only you can answer that for your dog.