The structure ["Neither ... nor ..."] requires parallel construction. In this case, both parts of the sentence should follow the same grammatical pattern. Furthermore, in standard written English, when "Neither" or "Nor" begins a clause, the subject and auxiliary verb are inverted. This is why "does he come" and "does he write" are correct. The inversion ("does he") is necessary for formal writing. Thus, the correct structure would be: "Neither does he [verb] nor does he [verb]." We observe that only Option D reflects this.