Yes, it can mean both things.
The literal meaning is "What does he have?" but as a figure of speech, it can mean "What's up with him? What's his problem?", presumably from something like "What problem/worry/etc. does he have?"
I'm not sure whether "denn" can be translated - literally, it's "What does he have, then?", but it doesn't really have a meaning of its own in this sentence, I think, so much as "flavouring" the sentence with a certain feeling. One of German's famous "Abtönpartikel" or modal particles. The "flavour" here might perhaps be translated as "..., I wonder?".
Have a look at http://www.duden.de/rechtschreibung/denn_verstaerkende_Partikel meaning 1a -- you will see that the word is not defined so much as described (i.e. it isn't given a meaning consisting of a single word or a short phrase, but instead there is a sentence saying when or how it is used):
drückt in Fragesätzen innere Anteilnahme, lebhaftes Interesse, Ungeduld, Zweifel o. Ä. des Sprechers oder der Sprecherin aus; überhaupt, eigentlich
or in English (my translation):
"in question sentences, expresses inner sympathy, vivid interest, impatience, doubts, or similar feelings of the speaker; at all, really"
What kind of flavour does it have?
"What's up with him?" is softer and concerned, like about your friend, "what's bothering him"
"What's his problem is more aggressive and angry, if you were frustrated by someone or a stranger was rude to you… you might say "what's his problem" but not "what's up with him"
In the Egyptian Arabic, not the standard Arabic, we are used to use some words to make the sentences more polite or soft or friendly, especially when asking for doing things, and it is also untranslatable. May be in English also some words for this purpose, to soften or sweeten the speech, especially when asking to do something or when giving commands.
This is really a silly question. It my be an idiomatic German expression, but if you don't know the correct connotation it does not help you to learn German. Direct translations are always meaningless. Idioms are interesting but not for beginners. The people who always give the (very helpful) explainations in the the discussions are obviously not beginners and probably don't need the course anymore. ( please don't leave !) I am learning to live with this because otherwise DL is very good.(and cheap!)
That's denn as a conjunction ( http://www.duden.de/rechtschreibung/denn_Konjunktion ) -- this sentence has denn as a particle ( http://www.duden.de/rechtschreibung/denn_verstaerkende_Partikel ) which doesn't have a meaning as such but expresses the speaker's feelings of surprise, impatience, compassion, curiosity, or something like that.
"What can he have? Whatever might be the problem with him? What does he have, then? Oh, why, what does he have? What about him - what does he have?" and others might be possible interpretations of the German sentence to try to convey some of the feelings it might evoke. (Note, I'm not suggesting you try them on Duolingo!)
The explanatory text for this lesson fails to provide english translations for the vocabulary, making it not very useful. There's no method for reporting issues with the explanatory sections, so sadly I'm mentioning it in the discussion area (which likely isn't going to help improve it)