What is the ruling on a wife's negligence that resulted in the theft of her possessions and house keys, knowing that she had given copies of the keys to her family without her husband's knowledge?

A Muslim should be discerning and take measures to protect his wealth. If negligence is a habit of your wife, then you both should advise each other and resolve problems calmly. As for the keys, this matter should not be a cause for dispute, and the default is to believe the wife unless proven otherwise, as she might have left them as a trust.