"Things like fever and coughing are part of the body's immune response,"
But not necessarily effective ones. If your lungs are irritated, you cough, whether or not there is something that needs to be expelled. Worse, extensive coughing can cause irritation, which leads to even more coughing.
If you are coughing up mucus, take an expectorant to decrease the viscosity of said mucus (e.g. bromhexine, acetylcysteine) and make the coughing more effective.
If it's a nonproductive dry cough, you should don't hesitate to take an anti-tussitive (cough suppressant), e.g. codeine, noscapine, dextromethorfan.
There's also a theory that common cold symptoms are an allergic reaction that's best suppressed with 1st-generation antihistamines. At least, runny nose and coughing don't expel the virus which is multiplying IN YOUR BODY CELLS.
Disclaimer: I am not an M.D..