The answer to your title question is general reference: 'mid-year exams' (or rarely, 'middle-of-year exams'), but 'end-of-year exams' or less commonly 'year-end exams'. As to why, perhaps it's because 'end-year' would seem to imply 1918 or 1945 say rather than December, but people feel uncomfortable using 'year-mid' as 'mid' as a noun is ... More @Wikipedia
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