Just a quick question. I'm going to enter in for a grade 6 Bb cornet exam. In the syllabus, it says trumpet candidates must do transposition. Does this mean that I do not have to do this as I play the cornet?

Yup!
I entered for Gr6 Trumpet, even though I play Flugel too, but thought the transposition could be fun . So stick to cornet if you don't want to tranpose!