On the Great Plains, nineteenth-century settlers used mud and grass to build their homes, doing it without timber and nails.
(A) settlers used mud and grass to build their homes, doing it without
(B) settlers used mud and grass to build their homes, did it without
(C) settlers used mud and grass to build their homes, making them while not having
(D) settlers used mud and grass to build their homes, making do without
(E) settlers’ homes were built of mud and grass, making do without
Here's my take on this question (I also apply the same knowledge as I take on any other SC Q's)
(A) says "doing it without". At this very point, you need to ask yourself what is "it" referring to? "It" cannot refer to homes as homes is a plural noun; therefore, the least the answer can do is put doing "them" without.
(B) repeats the same problem in (A), which is the pronoun problem, something that is deeply tested on all standardized tests when SC applies, and pronoun error is always and forever wrong.
(C) switches "it" to "them", for which you need to ask yourself again the question of what is "them"?" Is "them" referring to homes? Is "them" referring to settlers? If you cannot distinguish the referrent of that pronoun, the pronoun is ambiguous, which is (again) always and forever wrong.
(D) has the construct of "making do", which is correct. Even if you didn't know that, POE can bring you down to just D alone.
(E) switches the main focus of settlers to settlers' homes. The change is subtle, but in terms of the construction of the sentence, it distorts the meaning. Are the "homes" themselves making do without timber? That is impossible; it has to be that the settlers can build these houses w/out timbers.
After all the POE, the only one that can make any remote sense is (D).