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I am a bit confused with the formation of this sentence. My lines are:

What if I were a girl? Would I have been able to cope with the ordeal that women and girls live through every day?

But I am still confused about whether writing would I have been able to is the correct form.

Would it still be correct if I change that to

What if I were a girl? Could I have coped with the ordeal that women and girls live through every day?

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Both say the same. "to be able to" is just a more fancy expression for "can" and your grammar is correct for both. So it's just a matter of taste, which one you use. – Anke Apr 12 '13 at 7:46
So you mean to say that it won't be incorrect if I say "Would I have been able to cope with....?". Can I go with it? – Amit Pattnaik Apr 12 '13 at 7:49
Absolutely. Like I said, the meaning is the same and both are grammatically correct. – Anke Apr 12 '13 at 7:53
Thank you so much :)) – Amit Pattnaik Apr 12 '13 at 7:58
Please look at the many related can-vs-be-able questions, and possibly also the conditional-perfect questions as well. – tchrist Apr 12 '13 at 12:20

Would is better as it's expressing a hypothetical subjunctive. That's also why it's correctly "if I were a girl" and not the incorrect "if I was a girl".

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I was going to disagree with this, then I thought about it and realised you are right. Could is still possible, but in a way ambiguous because it could also be used in a realis case, whereas would definitely takes it into the irrealis (hypothetical). Here, since the previous sentence makes it clearly hypothetical either will do, but in general would is better. – Colin Fine Apr 12 '13 at 9:17
Thank you Colin :)) Yeah, I finally settled for "would". – Amit Pattnaik Apr 20 '13 at 9:40

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