Could herpes around the anus cause my stool to be bloody?

Yes. Can bleed and cause pain. Need to see a colorectal surgeon. Anal rectal bleeding is never normal.
Perhaps. If the herpetic lesions are open you may see blood on the surface of the stool or on the toilet paper. Blood in the body of the stool usually comes from a source higher in the colon which allows time for the blood to be incorporated in the feces.