I have given my view why I think each of them may be correct.
1) Why is the young Joe right handed but the old Joe left handed?
In example 1, I think there is no need to add "is" two times ie."young Joe is right handed but the old Joe is left handed?" and that is why a single "is" would suffice.
2) Why is the young Joe right handed but the old Joe is left handed?
Here I have taken only the first "is" ie from "..young Joe is right handed..." to make the question. So the second "is" should be retained in its proper place.
3) Why is it that the young Joe is right handed but the old Joe left handed?
I think this question is okay, but need to ascertain it.
4) Why is young Joe is right handed but the old Joe is left handed?
I have retained both "is"; just added an extra "is" to make a question.
Now I am confused on the correctness of these constructions. These constructions just flowed through my head, but a long divorce with Grammar books made me feel doubt regarding them. Please elaborate which of the above is correct and which is not?