What Causes Cough, Dark Circles Under Eyes And Lumps On Neck Of A Child?

My grandson is 15 months old. He has had a cough &runny nose about 10 weeks. Been to the doctor 3-4 times and they say it is viral or give him Amoxicilin and he still has it. He also has dark circles under his eyes & lumps on his neck just below his ears. They told my daughter to watch to a rash because he might have the mumps. My questions are could the circles be from allergies? And at 15 months old how dou care for mumps?