Having the right hand and arm being crippled by a sniper’s bullet during the First World War, Horace Pippin, a Black American painter, worked by holding the brush in his right hand and guiding its movements with his left.
Option Elimination -
(A) Having the right hand and arm being crippled by a sniper’s bullet during the First World War - "being" used as a modifier is wrong as it wrongly conveys that the right hand and arm are being crippled as we speak. This happened long ago, during the First World War, not now. Avoid the use of "being" as a modifier. Generally wrong on GMAT (except when used as a noun/noun phrase such as Being punctual is an essential trait in the business world/The philosophy course explores the nature of being (here the nature of being refers to the philosophical inquiry into existence and reality) or passive continuous verb such as - The car is being repaired by the mechanics).
(B) In spite of his right hand and arm being crippled by a sniper’s bullet during the First World War - I know this sounds pretty interesting as it presents the contrast, but we have the same issue of being.
(C) Because there had been a sniper’s bullet during the First World War that crippled his right hand and arm - "Because" introduces a subordinate adverbial clause that modifies the subject and a verb in the following clause, i.e., Horace Pippin worked...It gives a weird meaning: "Because there had been a bullet...., Horace Pippin worked. Not good.
(D) The right hand and arm being crippled by a sniper’s bullet during the First World War - the same issue of being.
(E) His right hand and arm crippled by a sniper’s bullet during the First World War - At first, this sounds very weird, but if we look closely, it's just a modifier that is modifying Horace Pippin. So here we have two modifiers for "Horace Pippin." One after and one before. This is usually done to avoid confusion because if put both modifiers after "Horace Pippin," it'll look bizarre: Horace Pippin, a Black American painter, his right hand and arm crippled by a sniper’s bullet during the First World War...Another issue with noun modifiers is that we must put them as close to the noun they modify, unlike the adverbials. So, the best solution here is to add one noun modifier before and put another one after the noun.