It sounds like the dog has kennel cough, a respiratory virus that is highly contagious. Almost all dogs from shelters will have it no matter how clean the shelter may be. Kennel cough is viral so it will sometimes get better on its own. Other times, it needs antibiotics. There is a short incubation period so you will often not see symptoms until the pet has been home for a week or longer.
Parasites can also cause coughing.
No matter what, you need to see the vet for antibiotics if the discharge from the eyes or nose is green/yellow. This indicates that there is an infection present.
If he's eating well and is in good spirits, you can wait and see if it resolves on its own.
Let me know if you still have questions.
