Strep throat doesn't cause respitory symptoms such as cough runny nose and trouble breathing right?

Possibly... Your throat infection could result in bronchospasm and airway inflammation which could lead to the symptoms you describe. So, see your doctor for evaluation as there are effective treatments.
Sure, it can do that. Occasionally, a patient shows up with stuffy nose symptoms, a mild runny nose, and tests positive for strep throat. Trouble breathing can be due to asthma-type symptoms, if the person is prone to getting such problems with any cold, flu, or strep symptoms. In more unusual cases, trouble breathing can be due to huge tonsils or a tonsillar abscess. Strep throat isn't always just a sore throat.
Mostly correct. Strep throat typically causes a sore throat, headache, abdominal pain, and fever. Occasionally, it can also be associated with a stuffy nose, but typically a cough.