EXAMPLE : At Shiprock, New Mexico, a perennially powerful girls ...

At Shiprock, New Mexico, a perennially powerful girls high school basketball team has become a path to college for some and a source of pride for a community where the household incomes of 49 percent of them are below the poverty level.

C. where 49 percent of the household incomes are
D. which has 49 percent of the household incomes

The difference is one of meaning, not of grammar (choice D is perfectly grammatical). What choice D actually means is that this one little community in N.M. has 49 percent of ALL the household incomes below the poverty level (i.e., in the entire country, or whatever is the whole universe of reference). That's unlikely, to say the least, and, more importantly, it distorts the clear meaning of the original sentence (you're never allowed to do so unless the original is nonsense).

The other thing that doesn't make sense in choice D is '49 percent of the incomes'. It's not 49 percent of a bunch of figures on an Excel spreadsheet; it's supposed to be 49 percent of the households (or of the households' incomes).

EXAMPLE : apprentice jewelers ...

After a few weeks’ experience, apprentice jewelers can usually begin to discriminate, though not with absolute certainty, genuine diamonds from imitation diamonds. 
(A) genuine diamonds from imitation diamonds 
(C) between genuine diamonds and imitation diamonds 

(A) genuine diamonds [ DO ] from imitation diamonds 

(C) between genuine diamonds and imitation diamonds [ Prepositional Phrase ]

DO so far from VERB [ discriminate ] is not preferred.

* i've also never seen a modifier stuck directly between a VERB and that verb's DIRECT OBJECT, as in (a) and (b); that is just ridiculously awkward. (c) fixes this problem because there's no direct object; the modifier is inserted before a prepositional phrase. that's common practice. 

Notes:

tips-verb-and-its-direct-object-should

EXAMPLE : living in a freestanding house ...

According to a recent poll, owning and living in a freestanding house on its own land is still a goal of a majority of young adults,/.../

B. as that for earlier generations

E. as it was of earlier generations

"It" correctly refers to "goal" ! E is best.

Notes

notes-it-vs-that

EXAMPLE : Embryonic mammalian circulatory system ...

A peculiar feature of the embryonic mammalian circulatory system is that in the area of the heart the cells adhere to one another, beating in unison and adopting specialized orientations exclusive of one another. 
(A) beating in unison and adopting 
(B) they beat in unison while adopting 
(C) beat in unison, and adopt 
(D) beating in unison yet adopting 
(E) even though they beat in unison and adopt 

a couple of comments: 

* you said 'the correct answer'. is (a) actually the official answer? i think (d) is the better choice, for the reasons outlined below. actually i like choice (f), my own version (see the end of this post), but (d) is the best of the choices given here. 
is this a real gmatprep problem? if so, i'm surprised the writers didn't compose a correct answer closer to the one i've labeled (f) below. 

-- 

(a) 
parallelism & mechanics are ok, making this the second-best of the choices here, but this one misses the point: the cells beat in unison, but they adopt orientations exclusive of one another. these facts are diametrically opposed to one another: the first talks about cooperative action, the second about completely independent action. that's a striking contrast, one that certainly deserves a CONTRAST TRANSITION such as 'but', 'yet', 'although', etc. 
not 'and'. 

(b) 
two ways to view the grammar here; either way it's wrong: 
- poor parallelism (your interpretation) 
- comma splice: you have an independent clause (i.e., something that could be a standalone sentence by itself), then a comma, then another independent clause. that's not allowed. 
note that this choice would produce a perfectly good sentence if the comma were replaced with a semicolon. 
also note that 'while' is an acceptable CONTRAST TRANSITION, although good transitions don't rescue a choice with bad grammar. 

(c) 
i see your point about the singular 'feature' referring to 3 different parallel components, but i don't think that's the issue here (one could conceive of the 3 different actions described as a single feature, if they're related closely enough). but that argument is moot, as this choice suffers from the same problem as (a): no CONTRAST TRANSITION to convey the intended meaning effectively. 

(d) 
all the grammatical advantages of choice (a), but with a proper CONTRAST TRANSITION. this one gets the thumbs up from me. 

(e) 
this isn't nonparallel, but the CONTRAST TRANSITION is in the wrong place: it's placed between 'adhere to one another' and 'beat in unison'. that's not the right contrast; the correct contrast is between the beating in unison (cooperative action) and adopting specialized orientations (individual action). 

EXAMPLE : A majority of the international journalists ...

 A majority of the international journalists surveyed view nuclear power stations as unsafe at present but that they will, or could, be made sufficiently safe in the future. 
(D) think that they will be or could 
(E) think the power stations would or could 

Ah, yes, I can see where you're coming from here: you don't like 'they' here. Neither do I, at least for GMAT purposes (meaning I wouldn't object to this sentence if I read it in a newspaper, but I know that the GMAT writers would ax it): 'they' could, technically, refer either to the power stations or to the journalists. 

BUT - choice E misuses 'would,' which is a past-tense construction. 
Correct: The journalists THOUGHT that the plants WOULD be made safe in the future. 
Correct: The journalists THINK that the plants WILL be made safe in the future. 
Incorrect: *The journalists THOUGHT that the plants WILL be made safe in the future.

Notes

notes-if-then-construct

EXAMPLE : Quetzalcoatlus ...

A huge flying reptile that died out with the dinosaurs some 65 million years ago, the Quetzalcoatlus had a wingspan of 36 feet, believed to be the largest flying creature the world has ever seen.

(A) believed to be

(B) and that is believed to be

(C) and it is believed to have been

A : 'believed to be' seems to modify wingspan and not Quetzalcoatlus.

B vs C

B is using a pointing pronoun to point Quetzalcoatlus which is never preferred on GMAT.

Note:

notes-it-vs-that

EXAMPLE : A collection of 38 poems by Phillis Wheatley ....

A collection of 38 poems by Phillis Wheatley, a slave, was published in the 1770s, the first book by a Black woman and  __________________________________________________

C. the second one only published by an American woman 
E. only the second published by an American woman

E.

C changes the meaning , it means that American Women published no other book , but this one. The E puts 

the meaning clearly saying that it is the 38 poems was the second to be published by an American Women.

Notes

notes-mind-only

EXAMPLE : The skill and the precision of the Anasazi ...

The skill and the precision of the Anasazi, ancient inhabitants of the Southwest, in measuring the movements of the Sun and Moon is evidenced not only at Chaco Canyon but at a number of other sites.

(B) in measuring the movements of the Sun and Moon are evidenced not only at
(E) to measure the movements of the Sun and Moon are evidenced not only at

The "to measure" is some how meaning ... now. The gerunds are generally preferred to express a well established facts [ B ]

Explanation