I've had an extremely sore throat for a week with some other mild respiratory symptoms. Is it worth a strep test?

Maybe. A week is a little long to have a persistent sore throat from a typical viral infection. If you are having a runny nose, cough, congestion, it is less likely to be strep throat but there are other possible explanations like seasonal allergies or infectious mononucleosis. I would go see your doctor for an evaluation.