Is it normal to have black stools a week after surgery?

Can't . Answer that question without knowing what the surgery as. Generally the first stools after colon surgery are mostly bile from the liver/gall bladder and can appear black. But black stools can be a warning of something more serious. Call your surgeon and make sure they are aware and be prepared to give details and possibly a visit to your surgeon or the er.
Not necessarily. If your stool is black and tarry, this can be a sign of blood in the stool (also known as melena). If your operation was involving some part of your gastrointestinal tract, then some residual clearing of blood with bowel movements is normal. However, you should mention this at your next appointment or contact your surgeon right away if it continues or worsens or becomes bright red.