My answer is
(A). It took me 01:08.
(A) In this version, "the ability to identify, without any visual cues, which note has been played" is an appositive of "absolute pitch". The meaning is clear. Keep for now.
(B) A reader cannot make sense of "identify which note was without any visual cues". At the very minimum, a comma should be added before "without". I wonder whether the omission is a typo.
(CDE) use a present participle phrase after "absolute pitch". When used as modifier after a comma, a present participle phrase typically modifies the subject of the preceding clause ("A characteristic" in this sentence) or modifies the verb in the preceding clause ("is" in this sentence).
(C) The logical subject for "identifying" should be "musicians", not "a characteristic".
(D) See above.
(E) It is logical to state that "a characteristic" permits somebody to do something, but "the musician" is not parallel with "musicians" in the main clause.