Must surgeons know, as a matter of routine, whether or not the
patients they are operating upon are hiv positive, irrespective of the
nature of the operation?

Surgeons and medical personnel operate on the rule of Universal
Precautions. All patients are treated as if they were potentially
infective. All blood and body fluid in a hospital or medical setting
is considered a biohazard. One never knows for sure, so we assume it
is all potentially dangerous.

Yes ! Most surgeons assume all of their patient HIV positive, upon
whom they are going to perform surgery.

Dear Segee, 

Yes, because surgeons must know the full medical file of the patient,
even things that seem "irrelevant" to the specific situation.

that is still personal protected information and you have the right to
refuse that information, if known, to anyone including health care
personell.  Likewise you can refuse to be tested for it.  HIV testing
is not a part of most pre-op procedures unless it is a
contraindication to the procedure itself (for example transplant
surgeries that will involve immune system suppression after the
surgery).  For more routine surgeries it is not a part of the
generally screening and the question of screening only comes up if an
accidental blood exposure to a health care worker occurs during the
procedure, such as a needle stick injury or an exposure to a mucous
membrane such as blood in an eye.  The doctors would still be required
to ask you for permission to check for the HIV virus and you would
still have the right to refuse testing for this.

As to whether an individual surgeon is willing to operate without
preoperative screening is a different matter.  It is however not a
universal rule that patients must be screened.