This is a great teaching question - we use it in class, actually. As neo pointed out, the opening clause is a modifier, not a complete sentence, so we don't need a coordinating conjunction to connect to the non-underlined portion. Also, some other reasons why the various answers are wrong:
(A) "Having" is the wrong tense - the sentence should be in past tense. "being" is used incorrectly.
(B) Meaning is wrong. He doesn't paint in this weird way "in spite of" being crippled, but because he was crippled. "being" is also used incorrectly.
(C) "Had been" is the wrong tense. For past perfect, the two events have to happen at distinct times. The bullet and the crippling are basically happening simultaneously.
(D) "The right hand and arm" is ambiguous. Whose hand and arm? Needs "his" instead of "the" to make sure we know we're talking about Horace's hand and arm. Also, "being" is used incorrectly.
(E) Sounds funny. But it's fine grammatically. This is how they'll get you to ignore / cross off a sentence that is grammatically correct - they make it sound funny but there's nothing actually wrong with it.
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