Red dots- Petichiae over the face can happen after a severe bout of cough
But I would suggest that you have him seen by your Pediatrician
as Petichiae can happen if platelet counts are low that can happen in many viral infections.
Good too know that his activity is normal.
Regarding his cough- if its more in early mornings and night, its likely that he has a wheeze too. He might need broncho-dilators in that case. Unless broncho-dilators are given his cough may not subside if its a wheeze.
For his runny nose he might need anti-histamines too.
So its better that you have him seen by your Pediatrician and get him treated appropriately.
Hope my answer helps you.
Wish your son a speedy recovery.