On the cough thing--he wasn't treated for heartworms, was he? I assume they would have told you if so. Most dogs in shelters in mosquito areas have heartworms. They treat them with poison that kills the worms, but the worms break apart and sometimes go into the lungs and cause the dog to cough. That's why you have to keep dogs on crate rest when they're recuperating from HW treatment--too much exercise causes more aspiration into the lungs.
As I said though, I would have thought that they would have told you. Could his records have gotten misplaced with all the confusion of the transfer?