"My wife's father has dark hair."
Translation:У отца моей жены тёмные волосы.
«У отца́» is more formal than «у па́пы». It's similar to the English words 'father' and 'Dad' (although I can't vouch the usage is completely the same).
In this sentence the first genitive is part of the Y+GEN expression and the second is simple possession. The idea of "having" is encapsulated in "у + gen" so if you start with "моей жены" first then she is the one who will have dark hair, not her father. The correct answer means "The father of my wife has...." Switching it as you write would mean. "My wife of father"... it doesn't really make sense.
You drop 'есть' when the sentence is about parts of the body. Maybe I am wrong, but I believe that if the hair were to be artificially added, like purchased and placed there, thus owned by the subject, maybe then they would say this sentence with 'есть', but not while it is a part of that person. Maybe "you have an artificial arm" would need 'есть'?
Here's an example that shows the usefulness of свой.
The sentence "He loves his wife" makes us feel good. But it's really ambiguous; its meaning could be sinister, if "he" and "his" don't have the same antecedent. (He and his could refer to different people. )
But the Russian "Он любит свою жену" is not ambiguous. The antecedent of свой is always the subject of the sentence.