I have to say I'm German, so I might lack the feel for English but to me the sentence "Does he eat THE dinner?" sounds totally wrong.
Is using the definite form in "Spiser han aftensmaden?" the usual way to ask "Is he having dinner (right now)?" in Danish or is it just one of Duo's awkward sentences to test the use of the definite article - and nobody would say this in Danish?
You are quite right. In Denmark, "aftensmaden" would normally be used for a specific dinner, whilst "aftensmad" would be used for dinner in general. In the UK, we would normally not use an article. In such cases where we would, we would usually say "your dinner" and not "the dinner".