Are brown (muddy colored) vaginal discharge and runny reddish stool common symptoms of an std?

Yes and no: Muddy and brown discharge may be due to std or non std infection and needs to be tested for that reddish stools is not a signof std but may be due to blood in stools and need to be evaluated by a physician or a gastroenterologist.

Possibly: Gonorrhea can cause a muddy discharge and if involves the rectum (proctitis), also cause a bloody diarrhea.