I don't have personal experience of these finer points, except that our DS got A* GCSE, and also did 'A' level music, fifteen years ago. I remember he had to know about the styles of Classical composers, and write passages in certain styles I think.
He was 'into' jazz much more than Classical, playing alto sax in school and county jazz bands. He also played keyboard, and went back to his early piano teacher to 'swot up' for the exams.
A proper understanding of the harmonic series, cycle of fifths etc is also useful, and I do mean UNDERSTANDING, and not just rote learning. I used to imagine 'harmony' was something dreamed up by early composers many hundreds of years ago, when in fact it is a natural result of the laws of PHYSICS of sound generation.
I would assume listening to as wide a range of music as possible, mainly Classical but also other genres, would be essential.
Maybe stillenacht can point people in the right direction for web sites on these topics? [PS: looked at your 'Profile', and concur your music tastes, with the exception of Amy: never did get on with her, Sorry!]