I've had a bad cough - coughing up mucus- for 2 weeks and now I also have a runny nose. Cough syrup helps a little bit. Is this just a cold?

Probable bronchitis. You probably have bronchitis. If you have only a low-grade fever, it is probably viral so antibiotics are not useful. If your symptoms are improving, you would need symptomatic relief only and you should see it resolve over the next few days. If your symptoms are worsening, see your doctor. You could also have residual bronchospasm best treated w/ inhalers that your doc can prescribe.
Probably not. If you have been coughing up colored mucus for 2 weeks and over the counter remedies aren't working, it's time to be checked out by a doctor for a possible lung infection.