A1 is fine A2 is poor grammar A3 is poor grammar (add "called" before his name, to fix it) B1 is fine B2 is poor grammar B3 is fine

#3a - comma after "writer". Otherwise, all perfect are perfect and have perfect grammar. The structure (in all six sentences) that places the profession "writer" is directly next to the name of a person is called "apposition". It is correct, and commonly used. Both sentences labeled #2 are correct in spite of the absence of an article. In both, the basic sentence is " This book was written by Neil Gaiman" which requires no article. When you add "famous English writer", no article is needed because apposition works without the addition of articles.

All of the sentences containing the English articles "a" and "the" are correct it is just a matter of personal preference. Native English speakers often drop the articles "a" and "the" in casual speaking, but know automatically when to include them at all other times. Both sentences numbered 2 from each/both blocks of three sentences are Bad "no article/articles sentences".

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