Freedmans survey showed that people living in small towns and rural

The difference here is not just about idiom or wordiness; it's about structure and meaning.

We are making a comparison, but what exactly are we comparing?

NO ___-ER THAN could set up a noun comparison or a verb comparison. Noun comparison: "I am no louder than you." Verb comparison: "I am no louder than you are" or "I laugh no louder than you do." (Notice that we use IS/ARE to refer back to another TO BE verb, while we can use DO to refer back to pretty much any other verb. Also notice that in the first of these, "louder" is an adjective, while in the second, "louder" is an adverb--we could have said "more loudly" to be clearer. The question uses "happier" as an adjective.)

Since A gives us "do people," it's setting up a verb comparison. The degree to which small town people consider themselves happy is no higher than the degree to which city people consider themselves happy.

NOT ANY ___-ER THAN sets up a noun comparison only. Why? It's the NOT. I can't say "I am not louder than you are." I am not what? Louder. Louder than what? You. "Are" doesn't fit anymore. The presence of NOT breaks up the sentence in such a way that we can't refer back to the previous verb. We have to compare nouns.

We might use this structure in B and C to convey a different meaning from A, and for all we know it's the correct one. (A doesn't have to have the right meaning.) People in small towns think they are NOT ANY HAPPIER THAN people in cities. So now we're not comparing how the two groups see themselves. We're looking at how one group (rural) compares itself to the other (city). This could work, but then the use of DO ruins the comparison. If we're comparing nouns, the verb DO is out of place.

This cuts B and C.

What about D? What happens when we change DO to ARE? Well, then we need a verb for ARE to refer back to, and it should also be a form of TO BE. (Notice that earlier we compared what I AM to what YOU ARE.) If we cut ARE, we'd avoid this problem and successfully convey the noun comparison that B and C were trying for. But we don't have that, so D fails.

(Also, the modifier switch in D is not great. "Living in X" is just a noun modifier--it describes the living place of the people in question. "Who are living" now has a verb in it, and it implies that this is happening in the present. That's not a crazy meaning, but it's not consistent with the earlier usage, and there's no reason for it.)

E? Again, ARE doesn't work without another similar TO BE verb to refer back to. So the meaning in A wins by default. Most of these choices are trying to make a different comparison than A is making, but none of them do so in a valid way! That's a bit unusual (often the correct meaning is repeated a bit more), but it can happen.