I'm sure you're worried about your little duckling. It sounds like he is in very caring hands.
I'd say that gapeworm is unlikely.
I'd say that stress alone would not be the problem either. Heat is a possibility - how hot would you say that the temperature is where you keep your duckling? If it is comfortable for you then it should be ok for him.
I would be looking for some other kind of underlying issue though. When you look at his nostrils and eyes do they appear to be perfectly clear or is there any kind of discharge at all? Is the area around his eye swollen at all? These things would indicate a respiratory infection.
Another possibility is that his nostrils are blocked (e.g. with dried food). Do they seem clear to you? Can you see all the way through them and out the other side? How often does he have access to water deep enough to put his whole head into? When he gets this water does he put his head in and clear out his nostrils by blowing air through them? To encourage him to clean his nostrils out, you can put finely chopped greens in a bowl of water, or put them in a bath for him.
Has he had an access to long grass? See if you can open his bill and check for any obstruction, including under his tongue.
What kind of other problems has he had? Has he been to see a vet?