I looked up "den Kopf kosten" and it was listed several times as "jdn den Kopf kosten" = "to cost somebody their job" (as well as "their head"). My translation was "That costs him his job in the end", which was marked wrong, but with no explanation. Should I flag this to add as a solution, or am I wrong? Is this expression not really used much so best not to add it?
Also, the given solution is "That will finally cost him his head". Shouldn't "That will cost..." be written as "Das wird ihm zuletzt den Kopf kosten"? The German given sentence seems to be present and the English answer seems to be future.
yes, it is used figuratively in the sense of 'lose your job/position/career/...'. But I guess you can use 'That will finally cost him his head' in that sense as well. So, I'm not sure whether it makes sense to add this as an additional solution. The present tense/future tense correspondance is fine IMHO. In German you can use the present more freely than in English to refer to future events.
I disagree, I think you only believe this because your native language (or some other language) makes you think this way. I think the simplest thing is to assume that the most obvious situation is at hand, and state that this is not so if otherwise. Which is what they do in German. And anyway it's not clear at all in English when you say "he lost his head" whether it's his own or some other male person's head. To be quite sure you'd have to add "own", which is awkward. More practical in this case are languages like Russian with a reflexive possessive pronoun that makes it easy to be very clear about whose head it is!