> Is it really likely that a 17-year-old A-student would be unfamiliar with the word "anachronism"?

Yes:-) It is likely that a surprisingly large proportion of all English speakers do not know the word. Not a majority perhaps, but a fair chunk nonetheless. So continue using it there Max!

As for Hamlet, the play's chronological setting is known to be imprecise but is, of course, generally considered to be the late medieval period. One fairly recent cinematic version (a pretty poor one) had Hamlet as the son of the owner of Denmark Corporation in New York, present day. Is that wrong? Considering the imprecision/universality of the play's text, I think such a setting choice can quite easily be justified even if the poor acting cannot.