This question seeks historical information about the cities of Sodom and Gomorrah, and their people. The Bible tells us nothing of historical value, beyond the plan to destroy the cities and the execution of that plan, which therefore places emphasis on any extra-biblical information available. It is not necessary for extra-biblical sources to confirm the biblical account of their destruction, although there ought to be mention of the two cities in the records of other civilisations. The Bible says that Sodom and Gomorrah stood out among all cities in the Near East because the people were so evil as to be beyond redemption, which at least ought to have elicited stories of disreputable cities that travellers should avoid. Unfortunately, every attempt to find extra-biblical references to the two cities, or even to interpret the same cities as being known by other names, has come to a dead end.
An argument from silence, reasonable in this instance, is that there is no extra-biblical record of the cities being destroyed in this way. Such an awe-inspiring event would surely have been recorded by other people, but with no record of it, there are now grounds for accepting that the destruction did not occur as portrayed in the Bible. Historically, the biblical account might have been a folk-memory of a natural disaster, perhaps similar to the volcano that destroyed Pompeii but on a far smaller scale.
Justification of God's actions
The question also requires in part that we judge God by human standards. In doing so, we have to see God's actions as ethical or, at least, fair, no matter how harsh. In answering the question, we can not say that God is beyond question, nor that he created us and is therefore entitled to deal with us as he wishes. If his justice is exhaustive and perfect, then the evidence will demonstrate this to be the case.
God's actions can be broken down into two parts: destroying the cities; and turning Lot's wife into a pillar of salt because she looked back. Fairness or otherwise in dealing with Lot's wife will shed light on the justification for God's treatment of these two cities.
Destruction of the cities
In Genesis 18:32, the Lord said that he would not destroy Sodom if as few as ten righteous persons could be found there. As a starting point, that seems a very reasonable promise. Notice that God does not say he knows how many righteous persons could be found there, so we need a story to establish just how thoroughly evil Sodom was.
The messengers went to Sodom and lodged with Lot and his wife and two daughters. The men of the entire town came and demanded to homosexually rape the two men (Genesis 19:5), proving how evil the place was. Notice that we simply assume that not ten men in the entire town stood back and refused to take part; notice also that all the women and children of the town have to be regarded as equally guilty as the men, or we must assume that women and children have no rights apart from their menfolk - otherwise God must at least spare them. Without proof that none of the town's men refused to take part in the attempted rape, and without proof that the women of the town were complicit, then God has not kept his word.
Genesis 19:24 says that Gomorrah was also destroyed in the same event. Was Gomorrah equally as evil as Sodom? From what we are told, we simply do not know. Justification has not been established by the biblical account.
Lot and his wife were told to flee the city of Sodom in case they were consumed and not to look back as they went (Genesis 19:17). There was no warning that any of the family would be punished in some way if they so much as looked back, and in any case it would have been unjust if they were threatened with reprisals for looking back. In Genesis 19:26:
But his wife looked back from behind him, and she became a pillar of salt.
If this is what happened to Lot's wife, it was entirely unjustified. We can not really say that it is either fair or just to destroy a wife and mother because God told her not to look back.