Learn English – Are the boys or the girl coming

'Headquarters' mean the main offices of any organization. Ideally, it is the central office and in most of the cases, it is one and the only. Said that, we find more use of 'the headquarters' as compared to 'a headquarters'.

Now, can it take the indefinite article 'a'? Yes, it can. How? Because 'headquarters' can serve as singular and plural both.

The company's headquarters is in Mumbai - is absolutely fine.

So, in the given sentence, 'a coalition headquarters in Qatar' is fine.

COCA shows several examples of 'a headquarters'. Yes, it's not incorrect.

I believe #3 will sound best and most natural to a native speaker. I would use two articles if I wanted to draw attention to some distinction between the chapters. For example:

The first chapter is difficult and the second chapter is difficult and long. In this sentence it would sound awkward to leave out the second "the".

It is not incorrect to use just one "the" as in your version #3, nor is it incorrect to use two. You may use the article twice, but to a native speaker it will sound stilted.

On the other hand, sometimes that is the effect you want. For example, if your neighbor is angry because you are playing loud music late at night, he might want to emphasize the distinct number of times he has asked you to turn down your music:

"I asked you politely the first time and the second time to turn down your music. The third time will not be polite."

But in normal everyday conversation you would just say,

"I asked him politely the first and second time, but the third time I punched him in the jaw."