Kohlberg attempted a tall feat indeed when he thought to quantify morality in individuals of different ages. He posed complex situational questions and asked his subjects to make a decision based on what they personally believed. It is an interesting approach because it allowed for the analysis of how different aged individuals typically view situations and what is "right" to them. Form his results he classified morality into three different groups. First Preconventional Morality, where right and wrong is based on what we are punished or rewarded for. I got yelled at for taking cookies from the cookie jar, so that is wrong. Second Conventional Morality, This is more on what is praised or rejected by society. Everyone in the family gets upset with me for taking the cookie out of the jar because there was not enough to share later. Finally Postconventional Morality, here right and wrong go beyonfd society and speaks form the essence of human rights. Everyone in the family deserves a cookie form the jar and should have an equal opportunity to get ones share.
His finding tell us that morality is not the same for everyone and that there isn't a consistent rule set that can be applied to every situation. Superman may make for a great story telling, but his adversaries are making it too easy on him. the real world is more complex and as we age and experience more complex situation our ability to analyze and judge become more refined.
This can be a complicated lesson to learn. It is not always easy to decide right from wrong even when we are present as there are so many other factors outside the actual incident it can be impossible to ever be truly sure. 
This brings up a powerful criticism with his analyses. With all of the different situational exposures we face and how those can be different across cultures, generations and even genders. It is unlikely that the people measured could be controlled for all of those factors given that there is inconsistencies even within those classifications. there is also the question of whether people would actual be honest in their answers or are experiencing a level of conventional morality in their answers playing to what they think is expected of them by those analyzing the tests. It would be hard to prevent this completely.
While there may be question about the data collected this does bring about some interesting question about how people develop their morality and the role empathy plays in that development. As we grow and experience are we collecting data so that we can better relate those around us to the point we have the ability to look past all the factors that surround a situation and truly tell right from wrong?
I like how you brought up possible flaws in Kohlberg's analyses. I definitely agree that a possible conflict was that people would be answering what they thought was the right answer. I also wonder whether the results he found would be different from today. Although a lot has changed, would his stages of moral development hold up in the current culture?