"What do you think" + Indirect question?

What do you think are the advantages and disadvantages of city life?

This is a perfectly grammatical and colloquial sentence. The analysis given in the question

"What do you think is followed by a direct question"

is wrong: are the advantages and disadvantages of city life is not a direct question. It's not a question at all; it's not even a complete clause -- no subject. It's just a verb phrase. In fact, it's the verb phrase in the question

• What are the advantages and disadvantages of city life?

which is what is being asked here. The do you think part is non-informational; how else would anybody answer except what they think?

The actual syntax is a result of the Wh-Question Formation rule, which can yank a Wh-word from almost anywhere in a sentence and stick it at the front of a question, after inverting the subject and first auxiliary of the question, like a Yes/No question.

This happens in stages, starting with the original non-question structure, with an unspecified argument that'll become the Wh-word. Most of these intermediate structures are ungrammatical (they're not finished) but I'll include them without question marks to illustrate the steps in the derivation. Here's the original structure that will result in the question:

• You think Unspec are the advantages and disadvantages of city life

To make a Wh-question out of this, you start by replacing the Unspec with the appropriate Wh-word:

• You think what are the advantages and disadvantages of city life

Then make a Yes/No question by inverting subject (you) and first auxiliary.

But, since You think has no auxiliary, Do-Support applies, and a new shiny auxiliary comes out of the slot and takes the tense morpheme, which is Zero in all cases so there's no change except adding do as an auxiliary:

• You do think what are the advantages and disadvantages of city life

and then inverting it with you:

• Do you think what are the advantages and disadvantages of city life

We're almost there now; the last step in Wh-Question Formation is to move the Wh-word what to the front, from wherever it was in the original:

• What do you think ___ are the the advantages and disadvantages of city life?

The hole in the sentence where the what came from is of course not audible. But it does mark a clause boundary; it's just that the subject of the clause has been moved up and out, giving the impression that what do you think is a constituent (it isn't, in this sentence) and that are the advantages ... is a question (as noted, it isn't).

Language is not a matter of words on a string like beads; there are constructions and they can change, by rule.

I just heard back from Editorial department in Cambridge University Press and this is what they respond:

"Sometimes in indirect questions where the subject is a very long phrase, we can reverse the order of the subject (the advantages and disadvantages of city life) and the verb (are) in order to make the question less unwieldy and more balanced."

In all honesty, I´m quite satisfied with the explanation.