Translation:After the dinner he usually sleeps.
Couldn't "After the dinner, he is usually asleep." also be a correct English translation for this sentence?
Technically, but it sounds very strange. Not because of "asleep" but because of "the dinner". The dinner would normally only refer to a specific special dinner, so "usually" wouldn't make sense. However, Duo probably cares about you using the verb "sleep". "Sleep" and "is asleep" don't FEEL interchangeable to me, but I don't have a good reason why.
why is it not "Efter aftensmaden HAN SOVER sædvanligvis"? or is that also correct? The word order feels funny to me
Indeed. The V2 rule (in most Germanic languages, English being the obvious exception here) states that the verb has to be the second 'phrase' in a sentence. Normally people say it's the second word in a phrase but fixed phrases like 'i dag' or 'i hvert fald' can stay in one piece at the beginning.
Han sover sædvanligvis. -- Sædvanligvis sover han.
Jeg begynder i dag. -- I dag begynder jeg.