A. The error is the use of 'because' in a separate sentence. The conjunction 'because' links two clauses within the same sentence. The correct sentence would be:
He went to the library because he wanted a book.
B. The error is the lack of a conjunction linking the two clauses, this is, a conjunction like 'because' is missing.
He went to the library because he wanted a book.
C, Went to the library is a case of run-on sentence because it has no subject.
The structure of this run-on sentence is: <*Subject missing> + <Main Verb> + <Preposition> + <Determiner> + <Noun/Prepositional Object>
D. The clause he went to the library expresses a consequence and he wanted a book expresses a cause, therefore the use of 'after,' expressing time, as a linking word is incorrect. Only conjunctions like 'because' or 'so' can be used in this case.
He went to the library because he wanted a book.
He wanted a book, so he went to the library.